let V be Z_Module; :: thesis: for W1, W2 being Submodule of V
for WW1, WW2 being Submodule of W1 + W2 st WW1 = W1 & WW2 = W2 holds
W1 /\ W2 = WW1 /\ WW2
let W1, W2 be Submodule of V; :: thesis: for WW1, WW2 being Submodule of W1 + W2 st WW1 = W1 & WW2 = W2 holds
W1 /\ W2 = WW1 /\ WW2
let WW1, WW2 be Submodule of W1 + W2; :: thesis: ( WW1 = W1 & WW2 = W2 implies W1 /\ W2 = WW1 /\ WW2 )
assume A1: ( WW1 = W1 & WW2 = W2 ) ; :: thesis: W1 /\ W2 = WW1 /\ WW2
A2: WW1 /\ WW2 is Submodule of V by ZMODUL01:42;
for x being object holds
( x in W1 /\ W2 iff x in WW1 /\ WW2 )

proof

let x be object ; :: thesis: ( x in W1 /\ W2 iff x in WW1 /\ WW2 )

hereby :: thesis: ( x in WW1 /\ WW2 implies x in W1 /\ W2 )

assume x in W1 /\ W2 ; :: thesis: x in WW1 /\ WW2
then ( x in WW1 & x in WW2 ) by A1, ZMODUL01:94;
hence x in WW1 /\ WW2 by ZMODUL01:94; :: thesis: verum

end;

assume x in WW1 /\ WW2 ; :: thesis: x in W1 /\ W2
then ( x in W1 & x in W2 ) by A1, ZMODUL01:94;
hence x in W1 /\ W2 by ZMODUL01:94; :: thesis: verum

end;

then for x being Vector of V holds
( x in W1 /\ W2 iff x in WW1 /\ WW2 ) ;
hence W1 /\ W2 = WW1 /\ WW2 by A2, ZMODUL01:46; :: thesis: verum