theorem
  a <> b & Collinear a,b,x & Collinear a,b,u & u <> x implies
  (are_orthogonal a,b,c,x iff (not Collinear a,b,c & right_angle c,x,u))
  proof
    assume that
A1: a <> b and
A2: Collinear a,b,x and
A3: Collinear a,b,u and
A4: u <> x;
    hereby
      assume
A5:   are_orthogonal a,b,c,x;
      then are_orthogonal a,b,x,c,x by A2,Satz8p15; then
A6:   a <> b & c <> x & are_orthogonal Line(a,b),x,Line(c,x);
A7:   u in Line(a,b) & c in Line(c,x) by A3,LemmaA1,GTARSKI3:83;
      then right_angle c,x,u by A6,Satz8p2; then
A8:   not Collinear c,x,u by A4,A5,Satz8p9;
A9:   not c in {y where y is POINT of S: Collinear u,x,y}
      proof
        assume c in {y where y is POINT of S: Collinear u,x,y};
        then ex y be POINT of S st c = y & Collinear u,x,y;
        hence contradiction by A8,GTARSKI3:45;
      end;
      Line(a,b) = Line(u,x)
      proof
        per cases;
        suppose u = a;
          hence thesis by A4,A1,GTARSKI3:82,A2,LemmaA1;
        end;
        suppose
A11:      u <> a; then
          Line(a,b) = Line(a,u) by A3,LemmaA1,A1,GTARSKI3:82;
          hence thesis by A4,A11,GTARSKI3:82,A2,LemmaA1;
        end;
      end; then
      not c in Line(a,b) by A9,GTARSKI3:def 10;
      hence not Collinear a,b,c & right_angle c,x,u by A7,A6,Satz8p2,LemmaA1;
    end;
    assume
A14: not Collinear a,b,c & right_angle c,x,u;
    now
      thus Line(a,b) is_line by A1,GTARSKI3:def 11;
      thus Line(c,x) is_line by A14,A2,GTARSKI3:def 11;
      thus x in Line(a,b) by A2,LemmaA1;
      thus x in Line(c,x) by GTARSKI3:83;
      thus ex u,v being POINT of S st u in Line(a,b) & v in Line(c,x) &
        u <> x & v <> x & right_angle u,x,v
      proof
        take u,c;
        thus u in Line(a,b) by A3,LemmaA1;
        thus c in Line(c,x) by GTARSKI3:83;
        thus u <> x by A4;
        thus c <> x by A14,A2;
        thus right_angle u,x,c by A14,Satz8p2;
      end;
    end;
    then are_orthogonal Line(a,b),Line(c,x) by Satz8p13;
    hence are_orthogonal a,b,c,x by A1,A14,A2;
  end;
