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 Th34:
  A is_line & not r in A & s in Plane(A,r) & not s in A implies
  Plane(A,r) = Plane(A,s)
  proof
    assume that
A1: A is_line and
A2: not r in A and
A3: s in Plane(A,r) and
A4: not s in A;
    consider r9 be POINT of S such that
A5: between r,A,r9 and
A6: Plane(A,r) = half-plane(A,r) \/ A \/ half-plane(A,r9) by A1,A2,Def10;
    consider s9 be POINT of S such that
A7: between s,A,s9 and
A8: Plane(A,s) = half-plane(A,s) \/ A \/ half-plane(A,s9) by A1,A4,Def10;
    s in half-plane(A,r) \/ A \/ half-plane(A,r9) &
       s in half-plane(A,r) \/ (A \/ half-plane(A,r9)) by A3,A6,XBOOLE_1:4;
    then (s in half-plane(A,r) \/ A or s in half-plane(A,r9)) &
      (s in half-plane(A,r) or s in A \/ half-plane(A,r9)) by XBOOLE_0:def 3;
    then per cases by A4,XBOOLE_0:def 3;
    suppose
A9:   s in half-plane(A,r);
        ex x be POINT of S st s = x & A out x, r by A9;
        then between r,A,s9 by A7,Th14;
        then  T1: between s9,A,r by GTARSKI3:14;
        between r9,A,r by A5,GTARSKI3:14;
        then A out s9,r9 by T1;
        then s9 in half-plane(A,r9);
        then half-plane(A,r9) = half-plane(A,s9) by A5,A7,Th23;
      hence thesis by A9,A1,A2,A4,Th23,A6,A8;
    end;
    suppose
A10:  s in half-plane(A,r9);
      then
A11:  half-plane(A,r9) = half-plane(A,s) by A4,A5,Th23;
      consider y be POINT of S such that
A12:  s = y and
A13:  A out y,r9 by A10;
      half-plane(A,r) = half-plane(A,s9)
      proof
        T1: between s9,A,s by A7,GTARSKI3:14;
        between r9,A,r & A out r9,y by A13,A5,GTARSKI3:14;
        then between y,A,r by Th14;
        then between r,A,s by A12,GTARSKI3:14;
        then A out s9,r by T1;
        then s9 in half-plane(A,r);
        hence thesis by A5,A7,Th23;
      end;
      hence thesis by A11,A6,A8,XBOOLE_1:4;
    end;
  end;
