:$1$ is represented by $\left[{1}\right]_\phi = \phi^0$. From there, the algorithm for Addition of 1 in Golden Mean Number System is run. {{begin-eqn}} {{eqn | n = 2       | l = 1 + \left[{1}\right]_\phi       | r = 1 + \left[{0 \cdotp 11}\right]_\phi       | c = Expansion }} {{eqn | r = \left[{1 \cdotp 11}\right]_\phi       | c =  }} {{eqn | r = \left[{10 \cdotp 01}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 3       | l = 1 + \left[{10 \cdotp 01}\right]_\phi       | r = \left[{11 \cdotp 01}\right]_\phi       | c =  }} {{eqn | r = \left[{100 \cdotp 01}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 4       | l = 1 + \left[{100 \cdotp 01}\right]_\phi       | r = \left[{101 \cdotp 01}\right]_\phi       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 5       | l = 1 + \left[{101 \cdotp 01}\right]_\phi       | r = 1 + \left[{101 \cdotp 0011}\right]_\phi       | c = Expansion }} {{eqn | r = 1 + \left[{100 \cdotp 1111}\right]_\phi       | c = Expansion }} {{eqn | r = \left[{101 \cdotp 1111}\right]_\phi       | c =  }} {{eqn | r = \left[{110 \cdotp 0111}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{1000 \cdotp 0111}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{1000 \cdotp 1001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 6       | l = 1 + \left[{1000 \cdotp 1001}\right]_\phi       | r = \left[{1001 \cdotp 1001}\right]_\phi       | c =  }} {{eqn | r = \left[{1010 \cdotp 0001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 7       | l = 1 + \left[{1010 \cdotp 0001}\right]_\phi       | r = \left[{1011 \cdotp 0001}\right]_\phi       | c =  }} {{eqn | r = \left[{1100 \cdotp 0001}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{10000 \cdotp 0001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 8       | l = 1 + \left[{10000 \cdotp 0001}\right]_\phi       | r = \left[{10001 \cdotp 0001}\right]_\phi       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 9       | l = 1 + \left[{10001 \cdotp 0001}\right]_\phi       | r = 1 + \left[{10000 \cdotp 1101}\right]_\phi       | c = Expansion }} {{eqn | r = \left[{10001 \cdotp 1101}\right]_\phi       | c =  }} {{eqn | r = \left[{10010 \cdotp 0101}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 10       | l = 1 + \left[{10010 \cdotp 0101}\right]_\phi       | r = \left[{10011 \cdotp 0101}\right]_\phi       | c =  }} {{eqn | r = \left[{10100 \cdotp 0101}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 11       | l = 1 + \left[{10100 \cdotp 0101}\right]_\phi       | r = \left[{10101 \cdotp 0101}\right]_\phi       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 12       | l = 1 + \left[{10101 \cdotp 0101}\right]_\phi       | r = 1 + \left[{10101 \cdotp 010011}\right]_\phi       | c = Expansion }} {{eqn | r = 1 + \left[{10101 \cdotp 001111}\right]_\phi       | c = Expansion }} {{eqn | r = 1 + \left[{10100 \cdotp 111111}\right]_\phi       | c = Expansion }} {{eqn | r = \left[{10101 \cdotp 111111}\right]_\phi       | c =  }} {{eqn | r = \left[{10110 \cdotp 011111}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{11000 \cdotp 011111}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{100000 \cdotp 011111}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{100000 \cdotp 100111}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{100000 \cdotp 101001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 13       | l = 1 + \left[{100000 \cdotp 101001}\right]_\phi       | r = \left[{100001 \cdotp 101001}\right]_\phi       | c =  }} {{eqn | r = \left[{100010 \cdotp 001001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 14       | l = 1 + \left[{100010 \cdotp 001001}\right]_\phi       | r = \left[{100011 \cdotp 001001}\right]_\phi       | c =  }} {{eqn | r = \left[{100100 \cdotp 001001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 15       | l = 1 + \left[{100100 \cdotp 001001}\right]_\phi       | r = \left[{100101 \cdotp 001001}\right]_\phi       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | n = 16       | l = 1 + \left[{100101 \cdotp 001001}\right]_\phi       | r = 1 + \left[{100100 \cdotp 111001}\right]_\phi       | c = Expansion }} {{eqn | r = \left[{100101 \cdotp 111001}\right]_\phi       | c =  }} {{eqn | r = \left[{100110 \cdotp 011001}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{101000 \cdotp 011001}\right]_\phi       | c = Simplification }} {{eqn | r = \left[{101000 \cdotp 100001}\right]_\phi       | c = Simplification }} {{end-eqn}} {{qed}}