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;

theorem
  (product" JumpParts InsCode goto(i1,R)).1 = NAT
proof
  dom product" JumpParts InsCode goto(i1,R) = {1} by Th25;
  then
A1: 1 in dom product" JumpParts InsCode goto(i1,R) by TARSKI:def 1;
  hereby
    let x be object;
    assume x in (product" JumpParts InsCode goto(i1,R)).1;
    then x in pi(JumpParts InsCode goto(i1,R),1) by A1,CARD_3:def 12;
    then consider g being Function such that
A2: g in JumpParts InsCode goto(i1,R) and
A3: x = g.1 by CARD_3:def 6;
    consider I being Instruction of SCM R such that
A4: g = JumpPart I and
A5: InsCode I = InsCode goto(i1,R) by A2;
    consider i2 such that
A6: I = goto(i2,R) by A5,Th17;
    g = <*i2*> by A4,A6;
    then x = i2 by A3;
    hence x in NAT by ORDINAL1:def 12;
  end;
  let x be object;
  assume x in NAT;
  then reconsider x as Element of NAT;
  JumpPart goto(x,R) = <*x*> & InsCode goto(i1,R) = InsCode goto(x,R);
  then
A7: <*x*> in JumpParts InsCode goto(i1,R);
  <*x*>.1 = x;
  then x in pi(JumpParts InsCode goto(i1,R),1) by A7,CARD_3:def 6;
  hence thesis by A1,CARD_3:def 12;
end;
