reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;

theorem Th2:
  for V being RealLinearSpace, y,z being Point of V
  for x being object holds x = (1-r)*y + r*z implies (x = y iff r = 0
  or y = z) & (x = z iff r = 1 or y = z)
proof
  let V be RealLinearSpace, y,z be Point of V;
  let x be object;
  assume
A1: x = (1-r)*y + r*z;
  hereby
    assume x = y;
    then 0.V = (1-r)*y + r*z - y by A1,RLVECT_1:5
      .= (1-r)*y - y + r*z by RLVECT_1:def 3
      .= (1-r)*y - 1 * y + r*z by RLVECT_1:def 8
      .= (1-r-1)*y + r*z by RLVECT_1:35
      .= r*z - r*y by RLVECT_1:79
      .= r*(z-y) by RLVECT_1:34;
    then r = 0 or z-y = 0.V by RLVECT_1:11;
    hence r = 0 or y = z by RLVECT_1:21;
  end;
  hereby
    assume
A2: r = 0 or y = z;
    per cases by A2;
    suppose
      r = 0;
      hence x = y + 0 * z by A1,RLVECT_1:def 8
        .= y + 0.V by RLVECT_1:10
        .= y by RLVECT_1:4;
    end;
    suppose
      z = y;
      hence x = (1-r+r)*y by A1,RLVECT_1:def 6
        .= y by RLVECT_1:def 8;
    end;
  end;
  hereby
    assume x = z;
    then 0.V = (1-r)*y + r*z - z by A1,RLVECT_1:5
      .= (1-r)*y + (r*z - z) by RLVECT_1:def 3
      .= (1-r)*y + (r*z + (-1)*z) by RLVECT_1:16
      .= (1-r)*y + (-1+r)*z by RLVECT_1:def 6
      .= (1-r)*y + (-(1-r))*z
      .= (1-r)*y - (1-r)*z by RLVECT_1:79
      .= (1-r)*(y-z) by RLVECT_1:34;
    then 1-r+r = 0+r or y-z = 0.V by RLVECT_1:11;
    hence r = 1 or y = z by RLVECT_1:21;
  end;
  assume
A3: r = 1 or y = z;
  per cases by A3;
  suppose
    r = 1;
    hence x = 0.V + 1 * z by A1,RLVECT_1:10
      .= 1 * z by RLVECT_1:4
      .= z by RLVECT_1:def 8;
  end;
  suppose
    y = z;
    hence x = (1-r+r)*z by A1,RLVECT_1:def 6
      .= z by RLVECT_1:def 8;
  end;
end;
