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 consider vv1, vv2 being Vector of (W1 + W2) such that

B1: ( vv1 in WW1 & vv2 in WW2 & x = vv1 + vv2 ) by VECTSP_5:1;

thus x in W1 + W2 by B1; :: 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 consider v1, v2 being Vector of V such that

B2: ( v1 in W1 & v2 in W2 & x = v1 + v2 ) by VECTSP_5:1;

( v1 in W1 + W2 & v2 in W1 + W2 ) by B2, VECTSP_5:2;

then reconsider vv1 = v1, vv2 = v2 as Vector of (W1 + W2) ;

( v1 in WW1 & v2 in WW2 & x = vv1 + vv2 ) by A1, B2, VECTSP_4:13;

hence x in WW1 + WW2 by VECTSP_5:1; :: thesis: verum

end;then consider v1, v2 being Vector of V such that

B2: ( v1 in W1 & v2 in W2 & x = v1 + v2 ) by VECTSP_5:1;

( v1 in W1 + W2 & v2 in W1 + W2 ) by B2, VECTSP_5:2;

then reconsider vv1 = v1, vv2 = v2 as Vector of (W1 + W2) ;

( v1 in WW1 & v2 in WW2 & x = vv1 + vv2 ) by A1, B2, VECTSP_4:13;

hence x in WW1 + WW2 by VECTSP_5:1; :: thesis: verum

then consider vv1, vv2 being Vector of (W1 + W2) such that

B1: ( vv1 in WW1 & vv2 in WW2 & x = vv1 + vv2 ) by VECTSP_5:1;

thus x in W1 + W2 by B1; :: thesis: verum

( x in W1 + W2 iff x in WW1 + WW2 ) ;

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