reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem Th36: :: (11.1)
  R = {} & X <> {} implies R.:^X = {}
proof
  assume that
A1: R = {} and
A2: X <> {};
  R.:^X c= {}
  proof
    let a be object;
    assume
A3: a in R.:^X;
    consider x being object such that
A4: x in X by A2,XBOOLE_0:def 1;
    a in Im(R,x) by A3,A4,Th24;
    hence thesis by A1;
  end;
  hence thesis;
end;
