set p1 = [0 ,0 ] .--> [1,0 ,1];
set p2 = [1,1] .--> [2,1,1];
set p3 = [2,0 ] .--> [3,0 ,(- 1)];
set p4 = [2,1] .--> [3,0 ,(- 1)];
set f = id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1;
set x1 = [0 ,0 ];
thus Zero_Tran . [0 ,0 ] = (((((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [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},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0 ] .--> [3,0 ,(- 1)])) . [0 ,0 ] by Th2
.= (((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) . [0 ,0 ] by Th2
.= ((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) . [0 ,0 ] by Th3
.= [1,0 ,1] by FUNCT_7:96 ; :: thesis: ( Zero_Tran . [1,1] = [2,1,1] & Zero_Tran . [2,0 ] = [3,0 ,(- 1)] & Zero_Tran . [2,1] = [3,0 ,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
set x = [1,1];
thus Zero_Tran . [1,1] = (((((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [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},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0 ] .--> [3,0 ,(- 1)])) . [1,1] by Th2
.= (((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) . [1,1] by Th3
.= [2,1,1] by FUNCT_7:96 ; :: thesis: ( Zero_Tran . [2,0 ] = [3,0 ,(- 1)] & Zero_Tran . [2,1] = [3,0 ,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
set x = [2,0 ];
thus Zero_Tran . [2,0 ] = (((((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0 ] .--> [3,0 ,(- 1)])) +* ([2,1] .--> [3,0 ,(- 1)])) . [2,0 ] by Th3
.= ((((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0 ] .--> [3,0 ,(- 1)])) . [2,0 ] by Th3
.= [3,0 ,(- 1)] by FUNCT_7:96 ; :: thesis: ( Zero_Tran . [2,1] = [3,0 ,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
set x = [2,1];
thus Zero_Tran . [2,1] = (((((id [:(SegM 4),{0 ,1}:],{(- 1),0 ,1},1) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0 ] .--> [3,0 ,(- 1)])) +* ([2,1] .--> [3,0 ,(- 1)])) . [2,1] by Th2
.= [3,0 ,(- 1)] by FUNCT_7:96 ; :: thesis: Zero_Tran . [3,1] = [4,1,(- 1)]
set x = [3,1];
thus Zero_Tran . [3,1] = [4,1,(- 1)] by FUNCT_7:96; :: thesis: verum