let R be Ring; :: thesis: for V being LeftMod of R

for W1, W2 being Subspace of V

for WW1, WW2 being Subspace of W1 + W2 st WW1 = W1 & WW2 = W2 holds

W1 /\ W2 = WW1 /\ WW2

let V be LeftMod of R; :: thesis: for W1, W2 being Subspace of V

for WW1, WW2 being Subspace of W1 + W2 st WW1 = W1 & WW2 = W2 holds

W1 /\ W2 = WW1 /\ WW2

let W1, W2 be Subspace of V; :: thesis: for WW1, WW2 being Subspace of W1 + W2 st WW1 = W1 & WW2 = W2 holds

W1 /\ W2 = WW1 /\ WW2

let WW1, WW2 be Subspace 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 Subspace of V by VECTSP_4:26;

for x being object holds

( x in W1 /\ W2 iff x in WW1 /\ WW2 )

( x in W1 /\ W2 iff x in WW1 /\ WW2 ) ;

hence W1 /\ W2 = WW1 /\ WW2 by A2, VECTSP_4:30; :: thesis: verum

for W1, W2 being Subspace of V

for WW1, WW2 being Subspace of W1 + W2 st WW1 = W1 & WW2 = W2 holds

W1 /\ W2 = WW1 /\ WW2

let V be LeftMod of R; :: thesis: for W1, W2 being Subspace of V

for WW1, WW2 being Subspace of W1 + W2 st WW1 = W1 & WW2 = W2 holds

W1 /\ W2 = WW1 /\ WW2

let W1, W2 be Subspace of V; :: thesis: for WW1, WW2 being Subspace of W1 + W2 st WW1 = W1 & WW2 = W2 holds

W1 /\ W2 = WW1 /\ WW2

let WW1, WW2 be Subspace 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 Subspace of V by VECTSP_4:26;

for x being object holds

( x in W1 /\ W2 iff x in WW1 /\ WW2 )

proof

then
for x being Vector of V holds
let x be object ; :: thesis: ( x in W1 /\ W2 iff x in WW1 /\ WW2 )

then ( x in W1 & x in W2 ) by A1, VECTSP_5:3;

hence x in W1 /\ W2 by VECTSP_5:3; :: thesis: verum

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

assume
x in WW1 /\ WW2
; :: thesis: x in W1 /\ W2assume
x in W1 /\ W2
; :: thesis: x in WW1 /\ WW2

then ( x in WW1 & x in WW2 ) by A1, VECTSP_5:3;

hence x in WW1 /\ WW2 by VECTSP_5:3; :: thesis: verum

end;then ( x in WW1 & x in WW2 ) by A1, VECTSP_5:3;

hence x in WW1 /\ WW2 by VECTSP_5:3; :: thesis: verum

then ( x in W1 & x in W2 ) by A1, VECTSP_5:3;

hence x in W1 /\ W2 by VECTSP_5:3; :: thesis: verum

( x in W1 /\ W2 iff x in WW1 /\ WW2 ) ;

hence W1 /\ W2 = WW1 /\ WW2 by A2, VECTSP_4:30; :: thesis: verum