reserve R for Ring,
  r for Element of R,
  a, b, d1, d2 for Data-Location of R,
  il, i1, i2 for Nat,
  I for Instruction of SCM R,
  s,s1, s2 for State of SCM R,
  T for InsType of the InstructionsF of SCM R,
  k for Nat;
reserve S for non trivial Ring,
  p, q for Data-Location of S,
  w for Element of S;

theorem Th36:
  IncAddr(goto(i1,R),k) = goto(i1 + k,R)
proof
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;
    assume
A5: x in dom JumpPart goto(i1,R);
    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;
  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 thesis by A8,A9,COMPOS_0:1;
end;
