reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th26:
  X is being_plane & Y is being_plane & M is being_line & N is
  being_line & M c= X & N c= X & M c= Y & N c= Y & M<>N implies X = Y
proof
  assume that
A1: X is being_plane and
A2: Y is being_plane and
A3: M is being_line and
A4: N is being_line and
A5: M c= X & N c= X and
A6: M c= Y & N c= Y and
A7: M<>N;
  consider a,b such that
A8: a in M and
A9: b in M and
A10: a<>b by A3,AFF_1:19;
A11: now
    given q such that
A12: q in M and
A13: q in N;
    consider p such that
A14: q<>p and
A15: p in N by A4,AFF_1:20;
A16: not p in M by A3,A4,A7,A12,A13,A14,A15,AFF_1:18;
A17: now
      assume b<>q;
      then not LIN b,q,p by A3,A9,A12,A16,Lm7;
      hence thesis by A1,A2,A5,A6,A9,A12,A15,Th25;
    end;
    now
      assume a<>q;
      then not LIN a,q,p by A3,A8,A12,A16,Lm7;
      hence thesis by A1,A2,A5,A6,A8,A12,A15,Th25;
    end;
    hence thesis by A10,A17;
  end;
  consider c,d such that
A18: c in N and
  d in N and
  c <>d by A4,AFF_1:19;
  now
    assume M // N;
    then not c in M by A7,A18,AFF_1:45;
    then not LIN a,b,c by A3,A8,A9,A10,Lm7;
    hence thesis by A1,A2,A5,A6,A8,A9,A18,Th25;
  end;
  hence thesis by A1,A3,A4,A5,A11,Th22;
end;
