A1: dom (id the Instructions of SCM+FSA ) = the Instructions of SCM+FSA by RELAT_1:71;
let i be Instruction of SCM+FSA ; :: thesis: ( ( i = halt SCM+FSA implies (Directed (Macro i)) . 0 = goto 2 ) & ( i <> halt SCM+FSA implies (Directed (Macro i)) . 0 = i ) )
A2: (Macro i) . 0 = i by SCMFSA6B:33;
0 in {0 ,1} by TARSKI:def 2;
then A3: 0 in dom (Macro i) by Th4;
A4: card (Macro i) = 2 by Th6;

hereby :: thesis: ( i <> halt SCM+FSA implies (Directed (Macro i)) . 0 = i )

A5: dom (id the Instructions of SCM+FSA ) = the Instructions of SCM+FSA by RELAT_1:71;
assume A6: i = halt SCM+FSA ; :: thesis: (Directed (Macro i)) . 0 = goto 2
dom ((halt SCM+FSA ) .--> (goto 2)) = {(halt SCM+FSA )} by FUNCOP_1:19;
then A7: i in dom ((halt SCM+FSA ) .--> (goto 2)) by A6, TARSKI:def 1;
rng (Macro i) c= the Instructions of SCM+FSA by RELAT_1:def 19;
hence (Directed (Macro i)) . 0 = (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto 2)) * (Macro i)) . 0 by A4, FUNCT_7:118
.= (((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto 2))) * (Macro i)) . 0 by A5, FUNCT_7:def 3
.= ((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto 2))) . i by A3, A2, FUNCT_1:23
.= ((halt SCM+FSA ) .--> (goto 2)) . i by A7, FUNCT_4:14
.= goto 2 by A6, FUNCOP_1:87 ;
:: thesis: verum

end;

A8: dom ((halt SCM+FSA ) .--> (goto 2)) = {(halt SCM+FSA )} by FUNCOP_1:19;
assume i <> halt SCM+FSA ; :: thesis: (Directed (Macro i)) . 0 = i
then A9: not i in dom ((halt SCM+FSA ) .--> (goto 2)) by A8, TARSKI:def 1;
rng (Macro i) c= the Instructions of SCM+FSA by RELAT_1:def 19;
hence (Directed (Macro i)) . 0 = (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto 2)) * (Macro i)) . 0 by A4, FUNCT_7:118
.= (((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto 2))) * (Macro i)) . 0 by A1, FUNCT_7:def 3
.= ((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto 2))) . i by A3, A2, FUNCT_1:23
.= (id the Instructions of SCM+FSA ) . i by A9, FUNCT_4:12
.= i by FUNCT_1:35 ;
:: thesis: verum