 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;

theorem Th04:
  r <> 0 & u is non zero implies r * u is non zero
  proof
    assume that
A1: r <> 0 and
A2: u is non zero;
    r * u <> 0.TOP-REAL 3
    proof
      assume
A3:   r * u = 0.TOP-REAL 3;
      u = 1 * u by RVSUM_1:52
       .= (1/r * r) * u by A1,XCMPLX_1:87
       .= 1/r * 0.TOP-REAL 3 by A3,RVSUM_1:49
       .= 0.TOP-REAL 3;
      hence contradiction by A2;
    end;
    hence thesis;
  end;
