reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,c9,x,y,z,p,q,q9 for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,a9,b,b9,c,c9 for POINT of S;
reserve S                 for non empty satisfying_Tarski-model
                                    TarskiGeometryStruct,
        A,A9              for Subset of S,
        x,y,z,a,b,c,c9,d,u,p,q,q9 for POINT of S;

theorem Satz8p13:
  are_orthogonal A,x,A9 iff A is_line & A9 is_line & x in A & x in A9 &
  (ex u,v being POINT of S st u in A & v in A9 & u <> x & v <> x &
  right_angle u,x,v)
  proof
    hereby
      assume
A1:   are_orthogonal A,x,A9;
      hence A is_line & A9 is_line & x in A & x in A9;
      consider p,q be POINT of S such that
A2:   p <> q and
A3:   A = Line(p,q) by A1,GTARSKI3:def 11;
      consider p9,q9 be POINT of S such that
A4:   p9 <> q9 and
A5:   A9 = Line(p9,q9) by A1,GTARSKI3:def 11;
      thus ex u,v being POINT of S st u in A & v in A9 & u <> x & v <> x &
        right_angle u,x,v
      proof
        per cases;
        suppose
A6:       x = p;
          take q;
A7:       q in A by A3,GTARSKI3:83;
          per cases;
          suppose
A8:         x = p9;
            take q9;
            thus thesis by A8,A4,A6,A2,A1,A5,A7,GTARSKI3:83;
          end;
          suppose
A9:         x <> p9;
            take p9;
            p9 in A9 by A5,GTARSKI3:83;
            hence thesis by A9,A6,A2,A3,A1,GTARSKI3:83;
          end;
        end;
        suppose
A10:      x <> p;
          take p;
A11:      p in A by A3,GTARSKI3:83;
          per cases;
          suppose
A12:        x = p9;
            take q9;
            q9 in A9 by A5,GTARSKI3:83;
            hence thesis by A12,A4,A10,A3,GTARSKI3:83,A1;
          end;
          suppose
A13:        x <> p9;
            take p9;
            thus thesis by A5,GTARSKI3:83,A13,A11,A10,A1;
          end;
        end;
      end;
    end;
    assume that
A14: A is_line and
A15: A9 is_line and
A16: x in A and
A17: x in A9 and
A18: ex u,v being POINT of S st u in A & v in A9 & u <> x & v <> x &
      right_angle u,x,v;
    consider u9,v9 be POINT of S such that
A19: u9 in A and
A20: v9 in A9 and
A21: u9 <> x and
A22: v9 <> x and
A23: right_angle u9,x,v9 by A18;
    now
      let u,v be POINT of S;
      assume that
A24:  u in A and
A25:  v in A9;
      Collinear x,u9,u by A14,A16,A19,A24,A21,GTARSKI3:90;
      then right_angle u,x,v9 by A23,A21,Satz8p3;
      then
A26:  right_angle v9,x,u by Satz8p2;
      Collinear x,v9,v by A15,A17,A20,A25,A21,GTARSKI3:90;
      then right_angle v,x,u by A22,A26,Satz8p3;
      hence right_angle u,x,v by Satz8p2;
    end;
    hence are_orthogonal A,x,A9 by A14,A15,A16,A17;
  end;
