
theorem Th28:
  for A being convex non empty Subset of TOP-REAL 3
  for n,u,v being Element of TOP-REAL 3 st
  (for w being Element of TOP-REAL 3 st w in A holds |( n,w )| <> 0) &
  u in A & v in A holds 0 < |( n,u )| * |( n,v )|
  proof
    let A be convex non empty Subset of TOP-REAL 3;
    let n,u,v be Element of TOP-REAL 3;
    assume that
A1: for w be Element of TOP-REAL 3 st w in A holds |( n,w )| <> 0 and
A2: u in A & v in A;
    set x = |( n,u )|,
        y = |( n,v )|;
    assume
A3: not 0 < |( n,u )| * |( n,v)|;
A4: x <> 0 & y <> 0 by A1,A2;
    then
A5: 0 < x / (x - y) < 1 by A3,BKMODEL3:1;
    reconsider l = x / (x - y) as non zero Real by A3,A4,BKMODEL3:1;
    reconsider w = l * v + (1 - l) * u as Element of TOP-REAL 3;
    x <> y
    proof
      assume x = y;
      then l = 0 by XCMPLX_1:49;
      hence contradiction;
    end;
    then
A6: 1 - l = - y / (x - y) by Th25;
    |( n,w )| = 0
    proof
      |( n , w )| = |( n, l * v )| + |( n, (1 - l) * u)| by EUCLID_2:26
                 .= l * |(n , v)| + |( n, (1 - l) * u)| by EUCLID_2:19
                 .= x / (x - y) * y + (-y) / (x - y) * x by A6,EUCLID_2:19;
      hence thesis;
    end;
    hence contradiction by A1,A2,A5,CONVEX1:def 2;
  end;
