let a be Data-Location ; :: thesis: for i1 being Element of NAT
for k being natural number holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
let i1 be Element of NAT ; :: thesis: for k being natural number holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
let k be natural number ; :: thesis: IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
X1: JumpPart (IncAddr ((a =0_goto i1),k)) = k + (JumpPart (a =0_goto i1)) by COMPOS_1:def 38;
then A1: dom (JumpPart (IncAddr ((a =0_goto i1),k))) = dom (JumpPart (a =0_goto i1)) by VALUED_1:def 2;
A2: dom (JumpPart (a =0_goto (i1 + k))) = dom <*(i1 + k)*> by RECDEF_2:def 2
.= Seg 1 by FINSEQ_1:55
.= dom <*i1*> by FINSEQ_1:55
.= dom (JumpPart (a =0_goto i1)) by RECDEF_2:def 2 ;
A3: for x being set 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 set ; :: 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 A4: 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 RECDEF_2:def 2;
then A6: x = 1 by FINSEQ_1:111;
set f = (JumpPart (a =0_goto i1)) . x;
A8: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = k + ((JumpPart (a =0_goto i1)) . x) by X1, A1, A4, VALUED_1:def 2;
(JumpPart (a =0_goto i1)) . x = <*i1*> . x by RECDEF_2:def 2
.= i1 by A6, FINSEQ_1:57 ;
hence (JumpPart (IncAddr ((a =0_goto i1),k))) . x = <*(i1 + k)*> . x by A6, A8, FINSEQ_1:57
.= (JumpPart (a =0_goto (i1 + k))) . x by RECDEF_2:def 2 ;
:: thesis: verum

end;

AA: AddressPart (IncAddr ((a =0_goto i1),k)) = AddressPart (a =0_goto i1) by COMPOS_1:def 38
.= <*a*> by RECDEF_2:def 3
.= AddressPart (a =0_goto (i1 + k)) by RECDEF_2:def 3 ;
BB: InsCode (IncAddr ((a =0_goto i1),k)) = InsCode (a =0_goto i1) by COMPOS_1:def 38
.= 7 by RECDEF_2:def 1
.= InsCode (a =0_goto (i1 + k)) by RECDEF_2:def 1 ;
JumpPart (IncAddr ((a =0_goto i1),k)) = JumpPart (a =0_goto (i1 + k)) by A1, A2, A3, FUNCT_1:9;
hence IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) by AA, BB, COMPOS_1:7; :: thesis: verum