let i1 be Element of NAT ; :: thesis: for k being natural number
for a being Int-Location holds IncAddr (a >0_goto i1),k = a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))
let k be natural number ; :: thesis: for a being Int-Location holds IncAddr (a >0_goto i1),k = a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))
let a be Int-Location ; :: thesis: IncAddr (a >0_goto i1),k = a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))
A1: InsCode (IncAddr (a >0_goto i1),k) = InsCode (a >0_goto i1) by AMISTD_2:def 14
.= 8 by SCMFSA_2:49
.= InsCode (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k))) by SCMFSA_2:49 ;
A2: dom (AddressPart (IncAddr (a >0_goto i1),k)) = dom (AddressPart (a >0_goto i1)) by AMISTD_2:def 14;
A3: for x being set st x in dom (AddressPart (a >0_goto i1)) holds
(AddressPart (IncAddr (a >0_goto i1),k)) . x = (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x

proof

let x be set ; :: thesis: ( x in dom (AddressPart (a >0_goto i1)) implies (AddressPart (IncAddr (a >0_goto i1),k)) . x = (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x )
assume A4: x in dom (AddressPart (a >0_goto i1)) ; :: thesis: (AddressPart (IncAddr (a >0_goto i1),k)) . x = (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x
then A5: x in dom <*i1,a*> by Th25;

per cases ( x = 1 or x = 2 ) by A5, Lm2;

suppose A6: x = 1 ; :: thesis: (AddressPart (IncAddr (a >0_goto i1),k)) . x = (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x

then (product" (AddressParts (InsCode (a >0_goto i1)))) . x = NAT by Th56;
then consider f being Element of NAT such that
A7: f = (AddressPart (a >0_goto i1)) . x and
A8: (AddressPart (IncAddr (a >0_goto i1),k)) . x = il. SCM+FSA ,(k + (locnum f,SCM+FSA )) by A4, AMISTD_2:def 14;
f = <*i1,a*> . x by A7, Th25
.= i1 by A6, FINSEQ_1:61 ;
hence (AddressPart (IncAddr (a >0_goto i1),k)) . x = <*(il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)),a*> . x by A6, A8, FINSEQ_1:61
.= (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x by Th25 ;
:: thesis: verum

end;

suppose A9: x = 2 ; :: thesis: (AddressPart (IncAddr (a >0_goto i1),k)) . x = (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x

then (product" (AddressParts (InsCode (a >0_goto i1)))) . x <> NAT by Th5, Th57;
hence (AddressPart (IncAddr (a >0_goto i1),k)) . x = (AddressPart (a >0_goto i1)) . x by A4, AMISTD_2:def 14
.= <*i1,a*> . x by Th25
.= a by A9, FINSEQ_1:61
.= <*(il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)),a*> . x by A9, FINSEQ_1:61
.= (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) . x by Th25 ;
:: thesis: verum

end;

end;

end;

dom (AddressPart (a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)))) = dom <*(il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)),a*> by Th25
.= Seg 2 by FINSEQ_3:29
.= dom <*i1,a*> by FINSEQ_3:29
.= dom (AddressPart (a >0_goto i1)) by Th25 ;
hence IncAddr (a >0_goto i1),k = a >0_goto (il. SCM+FSA ,((locnum i1,SCM+FSA ) + k)) by A1, A2, A3, FUNCT_1:9, MCART_1:95; :: thesis: verum