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
  A is_plane & not r in A implies
  space3(A,r) = {x where x is POINT of S :
     A out2 x,r or x in A or between2 r,A,x}
  proof
    assume that
A1: A is_plane and
A2: not r in A;
    consider r9 be POINT of S such that
A3: between2 r,A,r9 and
A4: space3(A,r) = half-space3(A,r) \/ A \/ half-space3(A,r9) by A1,A2,Def20;
    set P = {x where x is POINT of S: A out2 x,r or x in A or between2 r,A,x};
A5: space3(A,r) c= P
    proof
      let x be object;
      assume x in space3(A,r);
      then x in (half-space3(A,r) \/ A) or x in half-space3(A,r9)
        by A4,XBOOLE_0:def 3;
      then per cases by XBOOLE_0:def 3;
      suppose x in half-space3(A,r);
        then x in {x where x is POINT of S: A out2 x,r} by A1,A2,Def18;
        then ex y be POINT of S st x = y & A out2 y,r;
        hence thesis;
      end;
      suppose x in A;
        hence thesis;
      end;
      suppose
A6:     x in half-space3(A,r9);
        then reconsider y = x as POINT of S;
        x in {x where x is POINT of S: A out2 x,r9} by A3,A6,Def18;
        then consider z be POINT of S such that
A7:     x = z and
A8:     A out2 z,r9;
        between2 r9,A,r & A out2 r9,y by A7,A8,A3,GTARSKI3:14;
        then between2 y,A,r by Th74;
        then between2 r,A,y by GTARSKI3:14;
        hence thesis;
      end;
    end;
    P c= space3(A,r)
    proof
      let x be object;
      assume x in P;
      then consider y be POINT of S such that
A9:   y = x and
A10:  A out2 y,r or y in A or between2 r,A,y;
      per cases by A10;
      suppose A out2 y,r;
        then y in {x where x is POINT of S: A out2 x,r};
        then x in half-space3(A,r) by A9,A1,A2,Def18;
        then x in half-space3(A,r) \/ A by XBOOLE_0:def 3;
        hence thesis by A4,XBOOLE_0:def 3;
      end;
      suppose y in A;
        then x in half-space3(A,r) \/ A by A9,XBOOLE_0:def 3;
        hence thesis by A4,XBOOLE_0:def 3;
      end;
      suppose between2 r,A,y;
        then between2 y,A,r & between2 r9,A,r by A3,GTARSKI3:14;
        then A out2 y,r9;
        then x in {x where x is POINT of S: A out2 x,r9} by A9;
        then x in half-space3(A,r9) by A3,Def18;
        then x in A \/ half-space3(A,r9) by XBOOLE_0:def 3;
        then x in half-space3(A,r) \/ (A \/ half-space3(A,r9))
          by XBOOLE_0:def 3;
        hence thesis by A4,XBOOLE_1:4;
      end;
    end;
    hence thesis by A5;
  end;
