reserve a, b, d1, d2, d3, d4 for Int-Location,
  A, B for Data-Location,
  f, f1, f2, f3 for FinSeq-Location,
  il, i1, i2 for Nat,
  L for Nat,
  I for Instruction of SCM+FSA,
  s,s1,s2 for State of SCM+FSA,
  T for InsType of the InstructionsF of SCM+FSA,
  k for Nat;
reserve J,K for Element of Segm 13,
  b,b1,c,c1 for Element of SCM-Data-Loc,
  f,f1 for Element of SCM+FSA-Data*-Loc;
reserve a, b, d1, d2, d3, d4 for Int-Location,
  A, B for Data-Location,
  f, f1,
  f2, f3 for FinSeq-Location;

theorem
  (product" JumpParts InsCode (a =0_goto i1)).1 = NAT
proof
  dom product" JumpParts InsCode (a =0_goto i1) = {1} by Th24,SCMFSA_2:24;
  then
A1: 1 in dom product" JumpParts InsCode (a =0_goto i1) by TARSKI:def 1;
  hereby
    let x be object;
    assume x in (product" JumpParts InsCode (a =0_goto i1)).1;
    then x in pi(JumpParts InsCode (a =0_goto i1),1) by A1,CARD_3:def 12;
    then consider g being Function such that
A2: g in JumpParts InsCode (a =0_goto i1) and
A3: x = g.1 by CARD_3:def 6;
    consider I being Instruction of SCM+FSA such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (a =0_goto i1) by A2;
    consider i2, b such that
A6: I = b =0_goto i2 by A5,SCMFSA_2:24,36;
    g = <*i2*> qua FinSequence by A4,A6,Th15;
    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;
A7: <*x*>.1 = x;
  InsCode (a =0_goto i1) = 7 by SCMFSA_2:24;
  then
A8: InsCode (a =0_goto i1) = InsCode (a =0_goto x) by SCMFSA_2:24;
  JumpPart (a =0_goto x) = <*x*> by Th15;
  then <*x*> in JumpParts InsCode (a =0_goto i1) by A8;
  then x in pi(JumpParts InsCode (a =0_goto i1),1) by A7,CARD_3:def 6;
  hence thesis by A1,CARD_3:def 12;
end;
