let m be Element of NAT ; :: thesis:
let I be Program of SCM+FSA; :: thesis:
A1: dom (id the Instructions of SCM+FSA) = the Instructions of SCM+FSA by RELAT_1:71;
rng ((halt SCM+FSA) .--> (goto (card I))) = {(goto (card I))} by FUNCOP_1:14;
then A2: rng ((id the Instructions of SCM+FSA) +* ((halt SCM+FSA) .--> (goto (card I)))) c= (rng (id the Instructions of SCM+FSA)) \/ {(goto (card I))} by FUNCT_4:18;
rng (id the Instructions of SCM+FSA) = the Instructions of SCM+FSA by RELAT_1:71;
then A3: (rng (id the Instructions of SCM+FSA)) \/ {(goto (card I))} = the Instructions of SCM+FSA by ZFMISC_1:46;
dom ((halt SCM+FSA) .--> (goto (card I))) = {(halt SCM+FSA)} by FUNCOP_1:19;
then dom ((id the Instructions of SCM+FSA) +* ((halt SCM+FSA) .--> (goto (card I)))) = (dom (id the Instructions of SCM+FSA)) \/ {(halt SCM+FSA)} by FUNCT_4:def 1
.= the Instructions of SCM+FSA by A1, ZFMISC_1:46 ;
then reconsider f = (id the Instructions of SCM+FSA) +* ((halt SCM+FSA) .--> (goto (card I))) as Function of the Instructions of SCM+FSA, the Instructions of SCM+FSA by A2, A3, FUNCT_2:def 1, RELSET_1:11;
A4: IncAddr ((goto (card I)),m) = goto (m + (card I)) by SCMFSA_4:14;
dom (id the Instructions of SCM+FSA) = the Instructions of SCM+FSA by RELAT_1:71;
then A5: f = (id the Instructions of SCM+FSA) +* ((halt SCM+FSA),(goto (card I))) by FUNCT_7:def 3;
A6: ProgramPart I = I by RELAT_1:209;
A7: rng I c= the Instructions of SCM+FSA by RELAT_1:def 19;
ProgramPart (Directed I) = Directed I by RELAT_1:209
.= f * I by A5, A7, FUNCT_7:118 ;
hence Reloc ((ProgramPart (Directed I)),m) = IncAddr ((f * (Shift (I,m))),m) by VALUED_1:23
.= ((id the Instructions of SCM+FSA) +* ((halt SCM+FSA) .--> (goto (m + (card I))))) * (Reloc ((ProgramPart I),m)) by A6, A4, COMPOS_1:167 ;
:: thesis: