reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  for a,b being Real st a^2 + b^2 = 1 holds |[-b,a,0]| is non zero
  proof
    let a,b be Real;
    assume
A1: a^2 + b^2 = 1;
    assume |[-b,a,0]| is zero;
    then -b = 0 & a = 0 by EUCLID_5:4,FINSEQ_1:78;
    hence contradiction by A1;
  end;
