let i1 be Instruction-Location of SCM ; :: thesis: for k being natural number holds IncAddr (goto i1),k = goto (il. SCM ,((locnum i1) + k))
let k be natural number ; :: thesis: IncAddr (goto i1),k = goto (il. SCM ,((locnum i1) + k))
A1: InsCode (IncAddr (goto i1),k) = InsCode (goto i1) by AMISTD_2:def 14
.= 6 by MCART_1:7
.= InsCode (goto (il. SCM ,((locnum i1) + k))) by MCART_1:7 ;
A2: dom (AddressPart (IncAddr (goto i1),k)) = dom (AddressPart (goto i1)) by AMISTD_2:def 14;
A3: dom (AddressPart (goto (il. SCM ,((locnum i1) + k)))) = dom <*(il. SCM ,((locnum i1) + k))*> by MCART_1:def 2
.= Seg 1 by FINSEQ_1:def 8
.= dom <*i1*> by FINSEQ_1:def 8
.= dom (AddressPart (goto i1)) by MCART_1:def 2 ;
for x being set st x in dom (AddressPart (goto i1)) holds
(AddressPart (IncAddr (goto i1),k)) . x = (AddressPart (goto (il. SCM ,((locnum i1) + k)))) . x

proof

let x be set ; :: thesis: ( x in dom (AddressPart (goto i1)) implies (AddressPart (IncAddr (goto i1),k)) . x = (AddressPart (goto (il. SCM ,((locnum i1) + k)))) . x )
assume A4: x in dom (AddressPart (goto i1)) ; :: thesis: (AddressPart (IncAddr (goto i1),k)) . x = (AddressPart (goto (il. SCM ,((locnum i1) + k)))) . x
then x in dom <*i1*> by MCART_1:def 2;
then A5: x = 1 by Lm1;
then (product" (AddressParts (InsCode (goto i1)))) . x = NAT by Th35;
then consider f being Instruction-Location of SCM such that
A6: f = (AddressPart (goto i1)) . x and
A7: (AddressPart (IncAddr (goto i1),k)) . x = il. SCM ,(k + (locnum f)) by A4, AMISTD_2:def 14;
f = <*i1*> . x by A6, MCART_1:def 2
.= i1 by A5, FINSEQ_1:def 8 ;
hence (AddressPart (IncAddr (goto i1),k)) . x = <*(il. SCM ,((locnum i1) + k))*> . x by A5, A7, FINSEQ_1:def 8
.= (AddressPart (goto (il. SCM ,((locnum i1) + k)))) . x by MCART_1:def 2 ;
:: thesis: verum

end;

then AddressPart (IncAddr (goto i1),k) = AddressPart (goto (il. SCM ,((locnum i1) + k))) by A2, A3, FUNCT_1:9;
hence IncAddr (goto i1),k = goto (il. SCM ,((locnum i1) + k)) by A1, AMISTD_2:16; :: thesis: verum