theorem Satz8p21:
  a <> b implies ex p,t st are_orthogonal a,b,p,a & Collinear a,b,t &
    between c,t,p
  proof
    assume
A1: a <> b;
    per cases;
    suppose
      not Collinear a,b,c;
      hence thesis by Satz8p21p1;
    end;
    suppose
A2:   Collinear a,b,c;
      consider c9 being POINT of S such that
A3:   not Collinear a,b,c9 by A1,GTARSKI3:92;
      ex p,t st are_orthogonal a,b,p,a & Collinear a,b,t & between c9,t,p
        by A3,Satz8p21p1;
      hence thesis by A2,GTARSKI3:15;
    end;
  end;
