let i1 be Element of NAT ; :: thesis: for k being natural number holds IncAddr (goto i1),k = goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))
let k be natural number ; :: thesis: IncAddr (goto i1),k = goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))
A1: InsCode (IncAddr (goto i1),k) = InsCode (goto i1) by AMISTD_2:def 14
.= 6 by SCMFSA_2:47
.= InsCode (goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))) by SCMFSA_2:47 ;
A2: dom (AddressPart (IncAddr (goto i1),k)) = dom (AddressPart (goto i1)) by AMISTD_2:def 14;
A3: for x being set st x in dom (AddressPart (goto i1)) holds
(AddressPart (IncAddr (goto i1),k)) . x = (AddressPart (goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + 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+FSA ,((locnum i1,SCM+FSA ) + k)))) . x )
assume A4: x in dom (AddressPart (goto i1)) ; :: thesis: (AddressPart (IncAddr (goto i1),k)) . x = (AddressPart (goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x
then x in dom <*i1*> by Th23;
then A5: x = 1 by Lm1;
then (product" (AddressParts (InsCode (goto i1)))) . x = NAT by Th53;
then consider f being Element of NAT such that
A6: f = (AddressPart (goto i1)) . x and
A7: (AddressPart (IncAddr (goto i1),k)) . x = il. SCM+FSA ,(k + (locnum f,SCM+FSA )) by A4, AMISTD_2:def 14;
f = <*i1*> . x by A6, Th23
.= i1 by A5, FINSEQ_1:def 8 ;
hence (AddressPart (IncAddr (goto i1),k)) . x = <*(il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))*> . x by A5, A7, FINSEQ_1:def 8
.= (AddressPart (goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x by Th23 ;
:: thesis: verum

end;

dom (AddressPart (goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) = dom <*(il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))*> by Th23
.= Seg 1 by FINSEQ_1:def 8
.= dom <*i1*> by FINSEQ_1:def 8
.= dom (AddressPart (goto i1)) by Th23 ;
hence IncAddr (goto i1),k = goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)) by A1, A2, A3, FUNCT_1:9, MCART_1:95; :: thesis: verum