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 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;
