[(In 1,2),(In 0 ,2)] <> [1,1] by Lm2, ZFMISC_1:33;
then A1: not [(In 1,2),(In 0 ,2)] in dom (1,1 .--> 0 ) by Lm3, TARSKI:def 1;
A2: [(In 1,2),(In 0 ,2)] in dom (1,0 .--> 1) by Lm2, Lm4, TARSKI:def 1;
thus add_2 . (In 1,2),(In 0 ,2) = (((0 ,0 .--> 0 ) +* (0 ,1 .--> 1)) +* (1,0 .--> 1)) . (In 1,2),(In 0 ,2) by A1, FUNCT_4:12
.= (1,0 .--> 1) . (In 1,2),(In 0 ,2) by A2, FUNCT_4:14
.= In 1,2 by Lm2, FUNCOP_1:86 ; :: thesis: verum