set x = [1,1];
set x1 = [0,0];
set p1 = [0,0] .--> [1,0,1];
set p2 = [1,1] .--> [1,1,1];
set p3 = [1,0] .--> [2,1,1];
set p4 = [2,0] .--> [3,0,(- 1)];
set p5 = [2,1] .--> [3,0,(- 1)];
set p6 = [3,1] .--> [3,1,(- 1)];
set f = id ([:(SegM 4),{0,1}:],{(- 1),0,1},0);
thus Succ_Tran . [0,0] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [0,0] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [0,0] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [0,0] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [0,0] by Th2
.= (((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [0,0] by Th2
.= ((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) . [0,0] by Th3
.= [1,0,1] by FUNCT_7:94 ; :: thesis: ( Succ_Tran . [1,1] = [1,1,1] & Succ_Tran . [1,0] = [2,1,1] & Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
thus Succ_Tran . [1,1] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [1,1] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [1,1] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,1] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [1,1] by Th2
.= (((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [1,1] by Th3
.= [1,1,1] by FUNCT_7:94 ; :: thesis: ( Succ_Tran . [1,0] = [2,1,1] & Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [1,0];
thus Succ_Tran . [1,0] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [1,0] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [1,0] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,0] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [1,0] by Th2
.= [2,1,1] by FUNCT_7:94 ; :: thesis: ( Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [2,0];
thus Succ_Tran . [2,0] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [2,0] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [2,0] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [2,0] by Th3
.= [3,0,(- 1)] by FUNCT_7:94 ; :: thesis: ( Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [2,1];
thus Succ_Tran . [2,1] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [2,1] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [2,1] by Th2
.= [3,0,(- 1)] by FUNCT_7:94 ; :: thesis: ( Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [3,1];
thus Succ_Tran . [3,1] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [3,1] by Th3
.= [3,1,(- 1)] by FUNCT_7:94 ; :: thesis: Succ_Tran . [3,0] = [4,0,0]
thus Succ_Tran . [3,0] = [4,0,0] by FUNCT_7:94; :: thesis: verum