let x be object ; :: thesis: for R1, R2 being Relation holds Im ((R1 \ R2),x) = (Im (R1,x)) \ (Im (R2,x))
let R1, R2 be Relation; :: thesis: Im ((R1 \ R2),x) = (Im (R1,x)) \ (Im (R2,x))
thus Im ((R1 \ R2),x) c= (Im (R1,x)) \ (Im (R2,x)) :: according to XBOOLE_0:def 10 :: thesis: (Im (R1,x)) \ (Im (R2,x)) c= Im ((R1 \ R2),x)

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in Im ((R1 \ R2),x) or y in (Im (R1,x)) \ (Im (R2,x)) )
assume y in Im ((R1 \ R2),x) ; :: thesis: y in (Im (R1,x)) \ (Im (R2,x))
then A1: [x,y] in R1 \ R2 by Th9;
then A2: not [x,y] in R2 by XBOOLE_0:def 5;
A3: y in Im (R1,x) by A1, Th9;
not y in Im (R2,x) by A2, Th9;
hence y in (Im (R1,x)) \ (Im (R2,x)) by A3, XBOOLE_0:def 5; :: thesis: verum

end;

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in (Im (R1,x)) \ (Im (R2,x)) or y in Im ((R1 \ R2),x) )
assume A4: y in (Im (R1,x)) \ (Im (R2,x)) ; :: thesis: y in Im ((R1 \ R2),x)
then A5: not y in Im (R2,x) by XBOOLE_0:def 5;
A6: [x,y] in R1 by A4, Th9;
not [x,y] in R2 by A5, Th9;
then [x,y] in R1 \ R2 by A6, XBOOLE_0:def 5;
hence y in Im ((R1 \ R2),x) by Th9; :: thesis: verum