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
  E is_space3 implies ex a,b,c,d st not a,b,c,d are_coplanar &
  E = space3(a,b,c,d)
  proof
    assume E is_space3;
    then consider r be POINT of S, A be Subset of S such that
A1: A is_plane and
A2: not r in A and
A3: E = space3(A,r);
    consider a,b,c be POINT of S such that
A4: not Collinear a,b,c and
A5: A = Plane(a,b,c) by A1;
    take a,b,c,r;
    a in A & b in A & c in A by A4,A5,Th69;
    hence thesis by A5,A3,Def21,A1,A2,A4,Th87;
  end;
