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 :: (8)
  R1 c= R2 implies R1.:^X c= R2.:^X
proof
  assume
A1: R1 c= R2;
  let y be object;
  assume
A2: y in R1.:^X;
  then reconsider B as non empty set;
  reconsider y as Element of B by A2;
  for x being set st x in X holds y in Im(R2,x)
  proof
    let x be set;
    assume x in X;
    then y in Im(R1,x) by A2,Th25;
    then [x,y] in R1 by Th9;
    hence thesis by A1,Th9;
  end;
  hence thesis by Th25;
end;
