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 Th25:
  IncAddr(a>0_goto i1,k) = a>0_goto(i1+k)
proof
A1: JumpPart IncAddr(a>0_goto i1,k) = k + JumpPart (a>0_goto i1)
  by COMPOS_0:def 9;
  then
A2: dom JumpPart IncAddr(a>0_goto i1,k) = dom JumpPart (a>0_goto i1)
  by VALUED_1:def 2;
A3: dom JumpPart (a>0_goto(i1+k)) = dom <*i1 + k*>
    .= Seg 1 by FINSEQ_1:38
    .= dom <*i1*> by FINSEQ_1:38
    .= dom JumpPart (a>0_goto i1);
A4: for x being object 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 object;
    assume
A5: x in dom JumpPart (a>0_goto i1);
    then x in dom <*i1*>;
      then
A6:   x = 1 by FINSEQ_1:90;
     set f = (JumpPart (a>0_goto i1)).x;
A7:   (JumpPart IncAddr(a>0_goto i1,k)).x = k + f by A1,A2,A5,VALUED_1:def 2;
      thus
      (JumpPart IncAddr(a>0_goto i1,k)).x
         = <*i1+k*>.x by A6,A7
        .= (JumpPart (a>0_goto(i1+k))).x;
  end;
A8: AddressPart IncAddr(a>0_goto i1,k) = AddressPart (a>0_goto i1)
           by COMPOS_0:def 9
    .= <*a*>
    .= AddressPart (a>0_goto(i1+k));
A9:  InsCode IncAddr(a>0_goto i1,k) = InsCode (a>0_goto i1) by COMPOS_0:def 9
    .= 8
    .= InsCode (a>0_goto(i1+k));
   JumpPart IncAddr(a>0_goto i1,k) = JumpPart (a>0_goto(i1+k))
      by A2,A3,A4,FUNCT_1:2;
  hence thesis by A8,A9,COMPOS_0:1;
end;
