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 Th63:
  a,a,b,c are_coplanar
  proof
    per cases;
    suppose  not Collinear a,b,c;
      then consider E be Subset of S such that
      Plane(a,b,c) = E and
A1:   E is_plane and
A2:   a in E and
A3:   b in E and
A4:   c in E by Th49;
      take E;
      thus thesis by A1,A2,A3,A4;
    end;
    suppose Collinear a,b,c; then
A5:   c in Line(a,b);
      per cases;
      suppose
A6:     a <> b;
        then consider a9 be POINT of S such that
A7:     not Collinear a,b,a9 by GTARSKI3:92;
        consider E be Subset of S such that
        Plane(a,b,a9) = E and
A8:     E is_plane and
A9:     a in E and
A10:     b in E and a9 in E by A7,Th49;
        Line(a,b) c= E by A6,A8,A9,A10,Th46;
        hence thesis by A5,A8,A9,A10;
      end;
      suppose a = b;
        hence thesis by Th62;
      end;
    end;
  end;
