reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;
reserve S         for non empty satisfying_Lower_Dimension_Axiom
                                satisfying_Tarski-model
                                TarskiGeometryStruct,
        a,b,c,d,m,p,q,r,s,x for POINT of S,
        A,A9,E              for Subset of S;

theorem Th67:
  Collinear a,b,x & Collinear c,d,x &
  ((b <> x & c <> x) or (b <> x & d <> x) or (a <> x & c <> x) or
  (a <> x & d <> x)) implies a,b,c,d are_coplanar
  proof
    assume that
A1: Collinear a,b,x & Collinear c,d,x and
A2: (b <> x & c <> x) or (b <> x & d <> x) or (a <> x & c <> x) or
      (a <> x & d <> x);
    per cases by A2;
    suppose b <> x & c <> x;
      hence thesis by A1,Th66;
    end;
    suppose
A3:   b <> x & d <> x;
      Collinear d,c,x by A1,GTARSKI3:14;
      then a,b,d,c are_coplanar by A1,A3,Th66;
      hence thesis;
    end;
    suppose a <> x & c <> x;
      hence thesis by A1,Th64;
    end;
    suppose
A4:   a <> x & d <> x;
      Collinear b,a,x & Collinear d,c,x by A1,GTARSKI3:14;
      then b,a,d,c are_coplanar by A4,A1,Th65;
      hence thesis;
    end;
  end;
