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 Satz8p15:
  a <> b & Collinear a,b,x implies
  (are_orthogonal a,b,c,x iff are_orthogonal a,b,x,c,x)
  proof
    assume that
A1: a <> b and
A2: Collinear a,b,x;
    thus are_orthogonal a,b,c,x implies are_orthogonal a,b,x,c,x
      by Satz8p15a,A2;
    assume are_orthogonal a,b,x,c,x;
    then c <> x & are_orthogonal Line(a,b),Line(c,x);
    hence thesis by A1;
  end;
