 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th18:
  product <*REAL*> = REAL 1
  proof
A8: product <*REAL*> c= REAL 1
    proof
      let x be object;
      assume x in product <*REAL*>;
      then consider g be Function such that
A1:   x = g and
A2:   dom g = dom <*REAL*> and
A3:   for j be object st j in dom <*REAL*> holds
      g.j in (<*REAL*>).j by CARD_3:def 5;
A4:   dom g = Seg 1 by A2,FINSEQ_1:def 8;
      rng g c= REAL
      proof
        let u be object;
        assume u in rng g;
        then consider t be object such that
A5:     t in dom g and
A6:     u = g.t by FUNCT_1:def 3;
        t in {1} by A5,FINSEQ_1:2,A2,FINSEQ_1:def 8; then
A7:     t = 1 & 1 in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
        u = g.1 & 1 in dom <*REAL*> by A6,A7,FINSEQ_1:def 8;
        then g.1 in (<*REAL*>).1 by A3;
        hence thesis by A7,A6;
      end;
      then x in Funcs(Seg 1,REAL) by A1,FUNCT_2:def 2,A4;
      then x in 1-tuples_on REAL by FINSEQ_2:93;
      hence x in REAL 1 by EUCLID:def 1;
    end;
    REAL 1 c= product <*REAL*>
    proof
      let x be object;
      assume
A9:   x in REAL 1;
      then x in 1-tuples_on REAL by EUCLID:def 1; then
A10:  x in Funcs(Seg 1,REAL) by FINSEQ_2:93;
      reconsider g = x as Function by A9;
      now
        consider h be Function such that
A11:    h = g and
A12:    dom h = Seg 1 and
A13:    rng h c= REAL by A10,FUNCT_2:def 2;
        thus dom g = dom <*REAL*> by A11,A12,FINSEQ_1:def 8;
        hereby
          let j be object;
          assume
A14:      j in dom <*REAL*>; then
A15:      j in Seg 1 by FINSEQ_1:def 8;
          j in {1} by FINSEQ_1:2,A14,FINSEQ_1:def 8; then
A16:      j = 1 by TARSKI:def 1;
          g.j in REAL by A11,A13,A12,A15,FUNCT_1:3;
          hence g.j in <*REAL*>.j by A16;
        end;
        hence for j be object st j in dom <*REAL*> holds g.j in <*REAL*>.j;
      end;
      hence x in product <*REAL*> by CARD_3:def 5;
    end;
    hence thesis by A8;
  end;
