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

hereby :: thesis: ( i <> halt SCM+FSA implies (Directed (Macro i)) . (insloc 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)) . (insloc 0 ) = goto (insloc 2)
dom ((halt SCM+FSA ) .--> (goto (insloc 2))) = {(halt SCM+FSA )} by FUNCOP_1:19;
then A7: i in dom ((halt SCM+FSA ) .--> (goto (insloc 2))) by A6, TARSKI:def 1;
rng (Macro i) c= the Instructions of SCM+FSA by AMI_1:118;
hence (Directed (Macro i)) . (insloc 0 ) = (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto (insloc 2))) * (Macro i)) . (insloc 0 ) by A4, FUNCT_7:118
.= (((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto (insloc 2)))) * (Macro i)) . (insloc 0 ) by A5, FUNCT_7:def 3
.= ((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto (insloc 2)))) . i by A3, A2, FUNCT_1:23
.= ((halt SCM+FSA ) .--> (goto (insloc 2))) . i by A7, FUNCT_4:14
.= goto (insloc 2) by A6, FUNCOP_1:87 ;
:: thesis: verum

end;

A8: dom ((halt SCM+FSA ) .--> (goto (insloc 2))) = {(halt SCM+FSA )} by FUNCOP_1:19;
assume i <> halt SCM+FSA ; :: thesis: (Directed (Macro i)) . (insloc 0 ) = i
then A9: not i in dom ((halt SCM+FSA ) .--> (goto (insloc 2))) by A8, TARSKI:def 1;
rng (Macro i) c= the Instructions of SCM+FSA by AMI_1:118;
hence (Directed (Macro i)) . (insloc 0 ) = (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto (insloc 2))) * (Macro i)) . (insloc 0 ) by A4, FUNCT_7:118
.= (((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto (insloc 2)))) * (Macro i)) . (insloc 0 ) by A1, FUNCT_7:def 3
.= ((id the Instructions of SCM+FSA ) +* ((halt SCM+FSA ) .--> (goto (insloc 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