let K, L be Ring; :: thesis: for V being non empty ModuleStr over K

for W being non empty RightModStr over L

for x being Scalar of K

for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let V be non empty ModuleStr over K; :: thesis: for W being non empty RightModStr over L

for x being Scalar of K

for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let W be non empty RightModStr over L; :: thesis: for x being Scalar of K

for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let x be Scalar of K; :: thesis: for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let y be Scalar of L; :: thesis: for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let v be Vector of V; :: thesis: for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let w be Vector of W; :: thesis: ( V = opp W & x = y & v = w implies w * y = x * v )

assume A1: ( V = opp W & x = y & v = w ) ; :: thesis: w * y = x * v

set o = the rmult of W;

A2: opp the rmult of W = the lmult of (opp W) by Th10;

thus w * y = the rmult of W . (w,y) by VECTSP_2:def 7

.= x * v by A1, A2, FUNCT_4:def 8 ; :: thesis: verum

for W being non empty RightModStr over L

for x being Scalar of K

for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let V be non empty ModuleStr over K; :: thesis: for W being non empty RightModStr over L

for x being Scalar of K

for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let W be non empty RightModStr over L; :: thesis: for x being Scalar of K

for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let x be Scalar of K; :: thesis: for y being Scalar of L

for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let y be Scalar of L; :: thesis: for v being Vector of V

for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let v be Vector of V; :: thesis: for w being Vector of W st V = opp W & x = y & v = w holds

w * y = x * v

let w be Vector of W; :: thesis: ( V = opp W & x = y & v = w implies w * y = x * v )

assume A1: ( V = opp W & x = y & v = w ) ; :: thesis: w * y = x * v

set o = the rmult of W;

A2: opp the rmult of W = the lmult of (opp W) by Th10;

thus w * y = the rmult of W . (w,y) by VECTSP_2:def 7

.= x * v by A1, A2, FUNCT_4:def 8 ; :: thesis: verum