let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite standard-ins homogeneous regular J/A-independent COM-Struct of N
for I, J being Instruction of S st ex k being natural number st IncAddr (I,k) = IncAddr (J,k) holds
I = J
let S be non empty stored-program IC-Ins-separated definite standard-ins homogeneous regular J/A-independent COM-Struct of N; :: thesis: for I, J being Instruction of S st ex k being natural number st IncAddr (I,k) = IncAddr (J,k) holds
I = J
let I, J be Instruction of S; :: thesis: ( ex k being natural number st IncAddr (I,k) = IncAddr (J,k) implies I = J )
given k being natural number such that A1: IncAddr (I,k) = IncAddr (J,k) ; :: thesis: I = J
A2: InsCode I = InsCode (IncAddr (I,k)) by Def38
.= InsCode J by A1, Def38 ;
XX: AddressPart I = AddressPart (IncAddr (I,k)) by Def38
.= AddressPart J by A1, Def38 ;
X1: JumpPart (IncAddr (I,k)) = k + (JumpPart I) by Def38;
then Y1: dom (JumpPart I) = dom (JumpPart (IncAddr (I,k))) by VALUED_1:def 2;
X2: JumpPart (IncAddr (J,k)) = k + (JumpPart J) by Def38;
then Y2: dom (JumpPart J) = dom (JumpPart (IncAddr (J,k))) by VALUED_1:def 2;
A3: dom (JumpPart I) = dom (JumpPart J) by A2, Def4;
for x being set st x in dom (JumpPart I) holds
(JumpPart I) . x = (JumpPart J) . x

proof

let x be set ; :: thesis: ( x in dom (JumpPart I) implies (JumpPart I) . x = (JumpPart J) . x )
assume A4: x in dom (JumpPart I) ; :: thesis: (JumpPart I) . x = (JumpPart J) . x
A7: (JumpPart (IncAddr (I,k))) . x = k + ((JumpPart I) . x) by X1, Y1, A4, VALUED_1:def 2;
A9: (JumpPart (IncAddr (J,k))) . x = k + ((JumpPart J) . x) by X2, A3, A4, Y2, VALUED_1:def 2;
thus (JumpPart I) . x = (JumpPart J) . x by A1, A7, A9; :: thesis: verum

end;

then JumpPart I = JumpPart J by A3, FUNCT_1:9;
hence I = J by A2, XX, Th7; :: thesis: verum