reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;

theorem Th2:
  for r1,r2,r3,r4,r5,r6 being Real
  st r1 < r2 & r3 <= r4 & r5 < r6
  holds L[01](r1,r2,r3,r4)*L[01](r5,r6,r1,r2) = L[01](r5,r6,r3,r4)
  proof
    let r1,r2,r3,r4,r5,r6 be Real;
    set f1 = L[01](r1,r2,r3,r4);
    set f2 = L[01](r5,r6,r1,r2);
    set f3 = L[01](r5,r6,r3,r4);
    assume
A1: r1 < r2 & r3 <= r4 & r5 < r6;
A2: dom(f1*f2) = [#]Closed-Interval-TSpace(r5,r6) by FUNCT_2:def 1
    .= dom f3 by FUNCT_2:def 1;
    for x being object st x in dom(f1*f2) holds (f1*f2).x = f3.x
    proof
      let x be object;
      assume
A3:   x in dom(f1*f2);
      then
A4:   x in [#]Closed-Interval-TSpace(r5,r6);
      then
A5:   x in [.r5,r6.] by A1,TOPMETR:18;
      reconsider r = x as Real by A3;
A6:   r5 <= r & r <= r6 by A5,XXREAL_1:1;
A7:   rng f2 c= [#]Closed-Interval-TSpace(r1,r2) by RELAT_1:def 19;
      reconsider s = f2.x as Real;
      x in dom f2 by A4,FUNCT_2:def 1;
      then s in [#]Closed-Interval-TSpace(r1,r2) by A7,FUNCT_1:3;
      then s in [.r1,r2.] by A1,TOPMETR:18;
      then r1 <= s & s <= r2 by XXREAL_1:1;
      then
A8:  f1.s = (r4 - r3) / (r2 - r1) * (s - r1) + r3 by A1,BORSUK_6:35;
A9:  r2 - r1 <> 0 by A1;
A10:  (r4 - r3) / (r2 - r1) * s
      = (r4 - r3) / (r2 - r1) * ((r2 - r1) / (r6 - r5) * (r - r5) + r1)
      by A1,A6,BORSUK_6:35
      .= ((r4 - r3) / (r2 - r1)) * ((r2 - r1) / (r6 - r5)) * (r - r5) +
      (r4 - r3) / (r2 - r1) * r1
      .= ((r4 - r3) / (r6 - r5)) * ((r2 - r1) / (r2 - r1)) * (r - r5) +
      (r4 - r3) / (r2 - r1) * r1 by XCMPLX_1:85
      .= ((r4 - r3) / (r6 - r5)) * 1 * (r - r5) +
      (r4 - r3) / (r2 - r1) * r1 by A9,XCMPLX_1:60
      .= (r4 - r3) / (r6 - r5) * (r - r5) + (r4 - r3) / (r2 - r1) * r1;
      thus (f1*f2).x = f1.(f2.x) by A3,FUNCT_1:12
      .= f3.x by A10,A8,A1,A6,BORSUK_6:35;
    end;
    hence thesis by A2,FUNCT_1:2;
  end;
