theorem
  (product" JumpParts InsCode (a =0_goto k1)).1 = NAT
proof
 dom product" JumpParts InsCode (a =0_goto k1) = {1} by Th10;
  then
A1: 1 in dom product" JumpParts InsCode (a =0_goto k1) by TARSKI:def 1;
  hereby
    let x be object;
    assume x in (product" JumpParts InsCode (a =0_goto k1)).1;
    then x in pi(JumpParts InsCode (a =0_goto k1),1) by A1,CARD_3:def 12;
    then consider g being Function such that
A2: g in JumpParts InsCode (a =0_goto k1) and
A3: x = g.1 by CARD_3:def 6;
    consider I being Instruction of SCM such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (a =0_goto k1) by A2;
    InsCode I = 7 by A5;
    then consider i2, b such that
A6: I = b =0_goto i2 by AMI_5:14;
    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 (a =0_goto x) = <*x*> & InsCode (a =0_goto k1) = InsCode
  (a =0_goto x);
  then
A7: <*x*> in JumpParts InsCode (a =0_goto k1);
  <*x*>.1 = x;
  then x in pi(JumpParts InsCode (a =0_goto k1),1) by A7,CARD_3:def 6;
  hence thesis by A1,CARD_3:def 12;
end;
