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 Th68:
  a,b,c,d are_coplanar iff ex x st
  (Collinear a,b,x & Collinear c,d,x) or
  (Collinear a,c,x & Collinear b,d,x) or
  (Collinear a,d,x & Collinear b,c,x)
  proof
    per cases;
    suppose
A1:   a = b;
      Collinear a,a,c & Collinear c,d,c by GTARSKI4:4;
      hence thesis by A1,Th63;
    end;
    suppose
A2:   a = c;
      Collinear a,a,b & Collinear b,d,b & a,a,b,d are_coplanar
        by Th63,GTARSKI4:4;
      hence thesis by A2;
    end;
    suppose
A3:   a = d;
      Collinear a,a,b & Collinear b,c,b & a,a,b,c are_coplanar
        by Th63,GTARSKI4:4;
      hence thesis by A3;
    end;
    suppose
A4:   b = c;
      Collinear b,b,a & Collinear a,d,a & b,b,a,d are_coplanar
        by Th63,GTARSKI4:4;
      hence thesis by A4;
    end;
    suppose
A5:   b = d;
      Collinear b,b,a & Collinear a,c,a & b,b,a,c are_coplanar
        by Th63,GTARSKI4:4;
      hence thesis by A5;
    end;
    suppose
A6:   c = d;
      Collinear c,c,a & Collinear a,b,a & c,c,a,b are_coplanar
        by Th63,GTARSKI4:4;
      hence thesis by A6;
    end;
    suppose
A7:   a <> b & a <> c & a <> d & b <> c & b <> d & c <> d;
      hereby
        assume a,b,c,d are_coplanar;
        then consider E be Subset of S such that
A8:     E is_plane and
A9:     a in E and
A10:    b in E and
A11:    c in E and
A12:    d in E;
        per cases;
        suppose
A13:       Collinear a,b,c;
          Collinear c,c,d by GTARSKI3:46;
          then Collinear c,d,c;
          hence ex x st (Collinear a,b,x & Collinear c,d,x) or
            (Collinear a,c,x & Collinear b,d,x) or
            (Collinear a,d,x & Collinear b,c,x) by A13;
        end;
        suppose
A14:      not Collinear a,b,c;
          set A  = Line(a,b),
              A9 = Line(a,c);
A15:      E = Plane(a,b,c) by A14,Th47,A8,A9,A10,A11;
          then
A16:      E= Plane(A,c) by A14,Def11;
A17:      not Collinear a,c,b by A14,GTARSKI3:14;
A18:      E = Plane(a,c,b) by A14,A15,Th53
           .= Plane(A9,b) by A17,Def11;
          per cases;
          suppose d in A;
            then
A19:        ex y be POINT of S st d = y & Collinear a,b,y;
            Collinear d,d,c by GTARSKI3:46;
            then Collinear c,d,d;
            hence ex x st (Collinear a,b,x & Collinear c,d,x) or
              (Collinear a,c,x & Collinear b,d,x) or
              (Collinear a,d,x & Collinear b,c,x) by A19;
          end;
          suppose d in A9;
            then
A20:        ex y be POINT of S st d = y & Collinear a,c,y;
            Collinear d,d,b by GTARSKI3:46;
            then Collinear b,d,d;
            hence ex x st (Collinear a,b,x & Collinear c,d,x) or
              (Collinear a,c,x & Collinear b,d,x) or
              (Collinear a,d,x & Collinear b,c,x) by A20;
          end;
          suppose
A21:        not d in A & not d in A9;
A22:        a <> b by A14,GTARSKI3:46;
A23:        a <> c
            proof
              assume a = c;
              then Collinear a,c,b by GTARSKI3:46;
              hence contradiction by A14,GTARSKI3:14;
            end;
            per cases;
            suppose A out d,c & A9 out d,b;
              then a,b out d,c & a,c out d,b by A14,GTARSKI3:46,A23;
              then between b,Line(a,d),c by Th59;
              then
              consider t be POINT of S such that
A24:          t in Line(a,d) and
A25:          between b,t,c;
                t1:ex y be POINT of S st t = y & Collinear a,d,y by A24;
                  Collinear b,c,t by A25,GTARSKI4:7;
              hence ex x st (Collinear a,b,x & Collinear c,d,x) or
                (Collinear a,c,x & Collinear b,d,x) or
                (Collinear a,d,x & Collinear b,c,x) by t1;
            end;
            suppose
A26:          not A9 out d,b;
                 T1:A9 is_line by A23;
                not b in A9
                proof
                  assume b in A9;
                  then ex y be POINT of S st b = y & Collinear a,c,y;
                  hence contradiction by A14,GTARSKI3:14;
                end;
              then Plane(A9,b) = {x where x is POINT of S : A9 out x,b or
                x in A9 or between b,A9,x} by T1,Th32;
              then consider y be POINT of S such that
A27:          d = y and
A28:          A9 out y,b or y in A9 or between b,A9,y by A12,A18;
              consider t be POINT of S such that
A29:          t in A9 and
A30:          between b,t,d by A27,A28,A26,A21;
                t1:ex x be POINT of S st t = x & Collinear a,c,x by A29;
                Collinear b,d,t by A30,GTARSKI3:14;
              hence ex x st (Collinear a,b,x & Collinear c,d,x) or
                (Collinear a,c,x & Collinear b,d,x) or
                (Collinear a,d,x & Collinear b,c,x) by t1;
            end;
            suppose
A31:          not A out d,c;
                T1: A is_line by A22;
                 not c in A
                proof
                  assume c in A;
                  then ex y be POINT of S st c = y & Collinear a,b,y;
                  hence contradiction by A14;
                end;
              then Plane(A,c) = {x where x is POINT of S : A out x,c or
                x in A or between c,A,x} by T1,Th32;
              then consider y be POINT of S such that
A32:          d = y and
A33:          A out y,c or y in A or between c,A,y by A12,A16;
              consider t be POINT of S such that
A34:          t in A and
A35:          between c,t,d by A21,A31,A32,A33;
                t1:ex x be POINT of S st t = x & Collinear a,b,x by A34;
                Collinear c,d,t by A35,GTARSKI3:14;
              hence ex x st (Collinear a,b,x & Collinear c,d,x) or
                (Collinear a,c,x & Collinear b,d,x) or
                (Collinear a,d,x & Collinear b,c,x) by t1;
            end;
          end;
        end;
      end;
      assume ex x st (Collinear a,b,x & Collinear c,d,x) or
        (Collinear a,c,x & Collinear b,d,x) or
        (Collinear a,d,x & Collinear b,c,x);
      then consider x such that
A36:  (Collinear a,b,x & Collinear c,d,x) or
        (Collinear a,c,x & Collinear b,d,x) or
        (Collinear a,d,x & Collinear b,c,x);
      per cases by A36;
      suppose
A37:     Collinear a,b,x & Collinear c,d,x;
        (a <> x or b <> x) & (c <> x or d <> x) by A7;
        hence a,b,c,d are_coplanar by A37,Th67;
      end;
      suppose
A38:    Collinear a,c,x & Collinear b,d,x;
        (a <> x or c <> x) & (b <> x or d <> x) by A7;
        then a,c,b,d are_coplanar by A38,Th67;
        hence thesis;
      end;
      suppose
A39:    Collinear a,d,x & Collinear b,c,x;
        (a <> x or d <> x) & (b <> x or c <> x) by A7;
        then a,d,b,c are_coplanar by A39,Th67;
        hence thesis;
      end;
    end;
  end;
