let R be Ring; :: thesis: for i1, k being Nat holds IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R)
let i1, k be Nat; :: thesis: IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R)
A1: JumpPart (IncAddr ((goto (i1,R)),k)) = k + (JumpPart (goto (i1,R))) by COMPOS_0:def 9;
then A2: dom (JumpPart (IncAddr ((goto (i1,R)),k))) = dom (JumpPart (goto (i1,R))) by VALUED_1:def 2;
A3: dom (JumpPart (goto ((i1 + k),R))) = dom <*(i1 + k)*>
.= Seg 1 by FINSEQ_1:def 8
.= dom <*i1*> by FINSEQ_1:def 8
.= dom (JumpPart (goto (i1,R))) ;
A4: for x being object st x in dom (JumpPart (goto (i1,R))) holds
(JumpPart (IncAddr ((goto (i1,R)),k))) . x = (JumpPart (goto ((i1 + k),R))) . x

proof

let x be object ; :: thesis: ( x in dom (JumpPart (goto (i1,R))) implies (JumpPart (IncAddr ((goto (i1,R)),k))) . x = (JumpPart (goto ((i1 + k),R))) . x )
assume A5: x in dom (JumpPart (goto (i1,R))) ; :: thesis: (JumpPart (IncAddr ((goto (i1,R)),k))) . x = (JumpPart (goto ((i1 + k),R))) . x
then x in dom <*i1*> ;
then A6: x = 1 by Lm1;
reconsider f = (JumpPart (goto (i1,R))) . x as Element of NAT by ORDINAL1:def 12;
A7: (JumpPart (IncAddr ((goto (i1,R)),k))) . x = k + f by A5, A1, A2, VALUED_1:def 2;
thus (JumpPart (IncAddr ((goto (i1,R)),k))) . x = <*(i1 + k)*> . x by A6, A7
.= (JumpPart (goto ((i1 + k),R))) . x ; :: thesis: verum

end;

A8: InsCode (IncAddr ((goto (i1,R)),k)) = InsCode (goto (i1,R)) by COMPOS_0:def 9
.= 6
.= InsCode (goto ((i1 + k),R)) ;
A9: AddressPart (IncAddr ((goto (i1,R)),k)) = AddressPart (goto (i1,R)) by COMPOS_0:def 9
.= {}
.= AddressPart (goto ((i1 + k),R)) ;
JumpPart (IncAddr ((goto (i1,R)),k)) = JumpPart (goto ((i1 + k),R)) by A2, A3, A4, FUNCT_1:2;
hence IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R) by A8, A9, COMPOS_0:1; :: thesis: verum