let i1, k be Nat; :: thesis: for a being Int-Location holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
let a be Int-Location; :: thesis: IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
A1: InsCode (IncAddr ((a =0_goto i1),k)) = InsCode (a =0_goto i1) by COMPOS_0:def 9
.= 7 by SCMFSA_2:24
.= InsCode (a =0_goto (i1 + k)) by SCMFSA_2:24 ;
A2: a =0_goto i1 = [7,<*i1*>,<*a*>] by Th7;
A3: a =0_goto (i1 + k) = [7,<*(i1 + k)*>,<*a*>] by Th7;
A4: AddressPart (IncAddr ((a =0_goto i1),k)) = AddressPart (a =0_goto i1) by COMPOS_0:def 9
.= <*a*> by A2
.= AddressPart (a =0_goto (i1 + k)) by A3 ;
A5: JumpPart (IncAddr ((a =0_goto i1),k)) = k + (JumpPart (a =0_goto i1)) by COMPOS_0:def 9;
then A6: dom (JumpPart (IncAddr ((a =0_goto i1),k))) = dom (JumpPart (a =0_goto i1)) by VALUED_1:def 2;
A7: for x being object st x in dom (JumpPart (a =0_goto i1)) holds
(JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x

proof

let x be object ; :: thesis: ( x in dom (JumpPart (a =0_goto i1)) implies (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x )
assume A8: x in dom (JumpPart (a =0_goto i1)) ; :: thesis: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x
then x in dom <*i1*> by Th15;
then A9: x = 1 by FINSEQ_1:90;
set f = (JumpPart (a =0_goto i1)) . x;
A10: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = k + ((JumpPart (a =0_goto i1)) . x) by A6, A5, A8, VALUED_1:def 2;
(JumpPart (a =0_goto i1)) . x = <*i1*> . x by Th15
.= i1 by A9 ;
hence (JumpPart (IncAddr ((a =0_goto i1),k))) . x = <*(i1 + k)*> . x by A9, A10
.= (JumpPart (a =0_goto (i1 + k))) . x by Th15 ;
:: thesis: verum

end;

dom (JumpPart (a =0_goto (i1 + k))) = dom <*(i1 + k)*> by Th15
.= Seg 1 by FINSEQ_1:38
.= dom <*i1*> by FINSEQ_1:38
.= dom (JumpPart (a =0_goto i1)) by Th15 ;
then JumpPart (IncAddr ((a =0_goto i1),k)) = JumpPart (a =0_goto (i1 + k)) by A6, A7, FUNCT_1:2;
hence IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) by A1, A4, COMPOS_0:1; :: thesis: verum