set p1 = [0 ,0 ] .--> [1,0 ,1];
set p2 = [1,1] .--> [1,0 ,1];
set p3 = [1,0 ] .--> [2,0 ,1];
set p4 = [2,1] .--> [2,0 ,1];
set f = id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ;
set x1 = [0 ,0 ];
thus U3(n)Tran . [0 ,0 ] = (((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) +* ([2,1] .--> [2,0 ,1])) . [0 ,0 ] by Th2
.= ((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) . [0 ,0 ] by Th2
.= (((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) . [0 ,0 ] by Th2
.= ((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) . [0 ,0 ] by Th3
.= [1,0 ,1] by FUNCT_7:96 ; :: thesis: ( U3(n)Tran . [1,1] = [1,0 ,1] & U3(n)Tran . [1,0 ] = [2,0 ,1] & U3(n)Tran . [2,1] = [2,0 ,1] & U3(n)Tran . [2,0 ] = [3,0 ,0 ] )
set x = [1,1];
thus U3(n)Tran . [1,1] = (((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) +* ([2,1] .--> [2,0 ,1])) . [1,1] by Th2
.= ((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) . [1,1] by Th2
.= (((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) . [1,1] by Th3
.= [1,0 ,1] by FUNCT_7:96 ; :: thesis: ( U3(n)Tran . [1,0 ] = [2,0 ,1] & U3(n)Tran . [2,1] = [2,0 ,1] & U3(n)Tran . [2,0 ] = [3,0 ,0 ] )
set x = [1,0 ];
thus U3(n)Tran . [1,0 ] = (((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) +* ([2,1] .--> [2,0 ,1])) . [1,0 ] by Th2
.= ((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) . [1,0 ] by Th3
.= [2,0 ,1] by FUNCT_7:96 ; :: thesis: ( U3(n)Tran . [2,1] = [2,0 ,1] & U3(n)Tran . [2,0 ] = [3,0 ,0 ] )
set x = [2,1];
thus U3(n)Tran . [2,1] = (((((id [:(SegM 3),{0 ,1}:],{(- 1),0 ,1},0 ) +* ([0 ,0 ] .--> [1,0 ,1])) +* ([1,1] .--> [1,0 ,1])) +* ([1,0 ] .--> [2,0 ,1])) +* ([2,1] .--> [2,0 ,1])) . [2,1] by Th3
.= [2,0 ,1] by FUNCT_7:96 ; :: thesis: U3(n)Tran . [2,0 ] = [3,0 ,0 ]
set x = [2,0 ];
thus U3(n)Tran . [2,0 ] = [3,0 ,0 ] by FUNCT_7:96; :: thesis: verum