
theorem Th06:
  for u,v being Element of TOP-REAL 3 for a,b,c,d,e being Real st
  u = |[a,b,c]| & v = |[d,e,0]| & are_Prop u,v holds c = 0
  proof
    let u,v be Element of TOP-REAL 3;
    let a,b,c,d,e be Real;
    assume that
A1: u = |[a,b,c]| and
A2: v = |[d,e,0]| and
A3: are_Prop u,v;
    consider f be Real such that f <> 0 and
A5: |[a,b,c]| = f * |[d,e,0]| by A1,A2,A3,ANPROJ_1:1;
    f * |[d,e,0]| = |[ f * d,f * e, f * 0]| by EUCLID_5:8;
    then c = |[ f * d, f * e, f * 0]|`3 by A5,EUCLID_5:2
          .= f * 0 by EUCLID_5:2;
    hence thesis;
  end;
