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;
reserve S                 for non empty
                            satisfying_Lower_Dimension_Axiom
                            satisfying_Tarski-model
                            TarskiGeometryStruct,
        a,b,c,p,q,x,y,z,t for POINT of S;

theorem :: ExtPerp3:
  for a,b,c,d being POINT of S holds
    a <> b & b <> c & c <> d & a <> c & a <> d & b <> d &
      are_orthogonal b,a,a,c & Collinear a,c,d implies
        are_orthogonal b,a,a,d
  proof
    let a,b,c,d be POINT of S;
    assume
A1: a <> b & b <> c & c <> d & a <> c & a <> d & b <> d &
      are_orthogonal b,a,a,c & Collinear a,c,d; then
A3: d in Line(a,c) by LemmaA1;
    a in Line(a,c) by GTARSKI3:83;
    hence thesis by A1,A3,Prelim11;
  end;
