reserve a, b, d1, d2 for Data-Location,
  il, i1, i2 for Nat,
  I for Instruction of SCM,
  s, s1, s2 for State of SCM,
  T for InsType of the InstructionsF of SCM,
  k,k1 for Nat;

theorem Th23:
  IncAddr(SCM-goto i1,k) = SCM-goto(i1+k)
proof
A1: JumpPart IncAddr(SCM-goto i1,k) = k + JumpPart SCM-goto i1
   by COMPOS_0:def 9;
 then
A2: dom JumpPart IncAddr(SCM-goto i1,k) = dom JumpPart SCM-goto i1
   by VALUED_1:def 2;
A3: dom JumpPart SCM-goto(i1+k)
 = dom <*i1+k*>
    .= Seg 1 by FINSEQ_1:def 8
    .= dom <*i1*> by FINSEQ_1:def 8
    .= dom JumpPart SCM-goto i1;
A4: for x being object st x in dom JumpPart SCM-goto i1 holds (JumpPart
  IncAddr(SCM-goto i1,k)).x = (JumpPart SCM-goto(i1+k)).x
  proof
    let x be object;
    assume
A5: x in dom JumpPart SCM-goto i1;
    then x in dom <*i1*>;
    then
A6: x = 1 by FINSEQ_1:90;
    set f = (JumpPart SCM-goto i1).x;
A7: (JumpPart IncAddr(SCM-goto i1,k)).x = k + f by A5,A2,A1,VALUED_1:def 2;
    thus
    (JumpPart IncAddr(SCM-goto i1,k)).x = <*i1+k*>.x
     by A6,A7
      .= (JumpPart SCM-goto(i1+k)).x;
  end;
A8: AddressPart IncAddr(SCM-goto i1,k) = AddressPart SCM-goto i1
          by COMPOS_0:def 9
    .= {}
    .= AddressPart SCM-goto(i1+k);
A9: InsCode IncAddr(SCM-goto i1,k) = InsCode SCM-goto i1 by COMPOS_0:def 9
    .= 6
    .= InsCode SCM-goto(i1+k);
   JumpPart IncAddr(SCM-goto i1,k) = JumpPart SCM-goto(i1+k)
     by A2,A3,A4,FUNCT_1:2;
  hence thesis by A8,A9,COMPOS_0:1;
end;
