let k, loc be Element of NAT ; :: thesis: IncAddr ((goto loc),k) = goto (loc + k)
U: goto loc = [6,<*loc*>,{}] ;
V: goto (loc + k) = [6,<*(loc + k)*>,{}] ;
A: InsCode (IncAddr ((goto loc),k)) = InsCode (goto loc) by COMPOS_1:def 38
.= 6 by RECDEF_2:def 1
.= InsCode (goto (loc + k)) by RECDEF_2:def 1 ;
B: AddressPart (IncAddr ((goto loc),k)) = AddressPart (goto loc) by COMPOS_1:def 38
.= {} by RECDEF_2:def 3
.= AddressPart (goto (loc + k)) by RECDEF_2:def 3 ;
X1: JumpPart (IncAddr ((goto loc),k)) = k + (JumpPart (goto loc)) by COMPOS_1:def 38;
JumpPart (IncAddr ((goto loc),k)) = JumpPart (goto (loc + k))

proof

thus K: dom (JumpPart (IncAddr ((goto loc),k))) = dom (JumpPart (goto (loc + k))) by A, COMPOS_1:def 33; :: according to FUNCT_1:def 17 :: thesis: for b₁ being set holds
( not b₁ in proj1 (JumpPart (IncAddr ((goto loc),k))) or (JumpPart (IncAddr ((goto loc),k))) . b₁ = (JumpPart (goto (loc + k))) . b₁ )
A1: JumpPart (goto loc) = <*loc*> by RECDEF_2:def 2;
A2: JumpPart (goto (loc + k)) = <*(loc + k)*> by RECDEF_2:def 2;
let x be set ; :: thesis: ( not x in proj1 (JumpPart (IncAddr ((goto loc),k))) or (JumpPart (IncAddr ((goto loc),k))) . x = (JumpPart (goto (loc + k))) . x )
assume Z: x in dom (JumpPart (IncAddr ((goto loc),k))) ; :: thesis: (JumpPart (IncAddr ((goto loc),k))) . x = (JumpPart (goto (loc + k))) . x
dom <*(loc + k)*> = {1} by FINSEQ_1:4, FINSEQ_1:55;
then C: x = 1 by Z, K, A2, TARSKI:def 1;
thus (JumpPart (IncAddr ((goto loc),k))) . x = k + ((JumpPart (goto loc)) . x) by X1, Z, VALUED_1:def 2
.= loc + k by A1, C, FINSEQ_1:57
.= (JumpPart (goto (loc + k))) . x by A2, C, FINSEQ_1:57 ; :: thesis: verum

end;

hence IncAddr ((goto loc),k) = goto (loc + k) by A, B, COMPOS_1:7; :: thesis: verum