reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;
reserve i,j,k for Nat,
  t,r1,r2,s1,s2 for Real;
reserve D1 for non vertical non empty compact Subset of TOP-REAL 2,
  D2 for non horizontal non empty compact Subset of TOP-REAL 2,
  D for non vertical non horizontal non empty compact Subset of TOP-REAL 2;

theorem Th51:
  r1 <= r2 implies (t in [.r1,r2.] iff ex s1 st 0 <=s1 & s1 <= 1 &
  t = s1*r1 + (1-s1)*r2)
proof
  assume
A1: r1 <= r2;
  per cases by A1,XXREAL_0:1;
  suppose
A2: r1 = r2;
    then
A3: [.r1,r2.] = {r1} by XXREAL_1:17;
    hereby
      reconsider a19 = 1 as Real;
      assume
A4:   t in [.r1,r2.];
      take s = a19;
      thus 0 <=s & s <= 1;
      thus t = s*r1 + (1-s)*r2 by A3,A4,TARSKI:def 1;
    end;
    given s1 such that
    0 <=s1 and
    s1 <= 1 and
A5: t = s1*r1 + (1-s1)*r2;
    thus thesis by A2,A3,A5,TARSKI:def 1;
  end;
  suppose
A6: r1 < r2;
    hereby
      assume
A7:   t in [.r1,r2.];
       reconsider s1 = (r2-t)/(r2-r1) as Real;
      take s1;
A8:   r2 - r1 > 0 by A6,XREAL_1:50;
      t <= r2 by A7,XXREAL_1:1;
      then 0 <= r2-t by XREAL_1:48;
      hence 0 <=s1 by A8;
      r1 <= t by A7,XXREAL_1:1;
      then r2-t <= r2-r1 by XREAL_1:10;
      hence s1 <= 1 by A8,XREAL_1:185;
      thus t = t*(r2-r1)/(r2-r1) by A8,XCMPLX_1:89
        .= ((r2-t)*r1 + (t-r1)*r2)/(r2-r1)
        .= (r2-t)*r1/(r2-r1) + (t-r1)*r2/(r2-r1) by XCMPLX_1:62
        .= (r2-t)*r1/(r2-r1) + (t-r1)/(r2-r1)*r2 by XCMPLX_1:74
        .= ((r2-t)/(r2-r1))*r1 + (1*(r2-r1)-(r2-t))/(r2-r1)*r2 by XCMPLX_1:74
        .= s1*r1 + (1-s1)*r2 by A8,XCMPLX_1:127;
    end;
    given s1 such that
A9: 0 <=s1 and
A10: s1 <= 1 and
A11: t = s1*r1 + (1-s1)*r2;
A12: s1*r2 + (1-s1)*r2 = 1*r2;
    1 - s1 >= 0 by A10,XREAL_1:48;
    then
A13: (1-s1)*r1 <= (1-s1)*r2 by A6,XREAL_1:64;
    s1*r1 <= s1*r2 by A6,A9,XREAL_1:64;
    then
A14: t <= r2 by A11,A12,XREAL_1:6;
    s1*r1 + (1-s1)*r1 = 1*r1;
    then r1 <= t by A11,A13,XREAL_1:6;
    hence thesis by A14,XXREAL_1:1;
  end;
end;
