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 Th37:
  A,A9 Is p & r in A9 & r <> p implies A9 c= Plane(A,r)
  proof
    assume that
A1: A,A9 Is p and
A2: r in A9 and
A3: r <> p;
A4: A9 = Line(p,r) by A1,A2,A3,GTARSKI3:87;
A5: not r in A by A1,A2,A3,GTARSKI3:89;
    then
A6: Plane(A,r) = {x where x is POINT of S : A out x,r or x in A or
      between r,A,x} by A1,Th32;
    now
      let x be object;
      assume
A7:   x in A9;
      then consider y be POINT of S such that
A8:  x = y;
      consider z be POINT of S such that
A9:  y = z and
A10:  Collinear p,r,z by A4,A7,A8;
      per cases;
      suppose
A11:    x = p;
        A c= Plane(A,r) by A1,A5,Th31;
        hence x in Plane(A,r) by A11,A1;
      end;
      suppose
A12:    x <> p;
        per cases;
        suppose between r,p,z;
          then between r,A,z by A1,A2,A3,A9,A8,A7,A12,GTARSKI3:89;
          hence x in Plane(A,r) by A8,A9,A6;
        end;
        suppose
A13:      not between r,p,z;
          Collinear r,p,z by A10,GTARSKI3:14;
          then
A14:      p out r,z by A13,GTARSKI3:73;
A15:      not reflection(p,r) in A
          proof
            assume
A16:        reflection(p,r) in A;
            A is_line & A9 is_line & reflection(p,r) in A9 by A1,A2,Th36;
            then Middle r,p,p by A1,A16,GTARSKI3:89,def 13;
            hence contradiction by A3,GTARSKI1:def 7;
          end;
              between z,p,reflection(p,r)
              proof
                per cases by A14;
                suppose between p,r,z;
                  then
A17:              between z,r,p by GTARSKI3:14;
                  Middle r,p,reflection(p,r) by GTARSKI3:def 13;
                  hence thesis by A3,A17,GTARSKI3:19;
                end;
                suppose between p,z,r;
                  then
A18:              between r,z,p by GTARSKI3:14;
                  Middle r,p,reflection(p,r) by GTARSKI3:def 13;
                  hence thesis by A18,GTARSKI3:18;
                end;
            end;
            then U1: between z,A,reflection(p,r)
              by A9,A8,A7,A12,GTARSKI3:89,A15,A1;
              Middle r,p,reflection(p,r) by GTARSKI3:def 13;
            then between r,A,reflection(p,r)
             by A1,A2,A3,GTARSKI3:89,A15;
          then A out z,r by U1;
          hence x in Plane(A,r) by A9,A8,A6;
        end;
      end;
    end;
    hence thesis;
  end;
