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

theorem Th24:
  for a,b,x being Real st a <= x <= b holds
    <*x*> in product <*[.a,b.]*>
  proof
    let a,b,x be Real;
    assume
A1: a <= x <= b;
    reconsider P = <*x*> as Point of Euclid 1 by Th7;
    set f = <*[.a,b.]*>;
A2: dom f = Seg 1 by FINSEQ_1:def 8;
    ex g be Function st g = P & dom g = dom <*[.a,b.]*> &
    for y be object st y in dom <*[.a,b.]*> holds
    g.y in (<*[.a,b.]*>).y
    proof
      reconsider g = P as Function;
      now
        take g;
        thus g = P;
        thus dom g = dom f by A2,FINSEQ_1:def 8;
        hereby
          let y be object;
          assume y in dom f;
          then y in {1} by FINSEQ_1:def 8,FINSEQ_1:2; then
A3:       y = 1 by TARSKI:def 1;
          thus g.y in f.y by A3,A1,XXREAL_1:1;
        end;
      end;
      hence thesis;
    end;
    hence thesis by CARD_3:def 5;
  end;
