let R1, R2 be Ring; :: thesis: for x, y being Scalar of R1
for p, q being Scalar of R2
for v, w being Vector of (BiModule R1,R2) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p )
set G = BiModule R1,R2;
set a = {} ;
let x, y be Scalar of R1; :: thesis: for p, q being Scalar of R2
for v, w being Vector of (BiModule R1,R2) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p )
let p, q be Scalar of R2; :: thesis: for v, w being Vector of (BiModule R1,R2) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p )
let v, w be Vector of (BiModule R1,R2); :: thesis: ( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p )
( x * (v + w) = {} & (x * v) + (x * w) = {} & (x + y) * v = {} & (x * v) + (y * v) = {} & (x * y) * v = {} & x * (y * v) = {} & (1_ R1) * v = {} & v = {} ) by CARD_1:87, TARSKI:def 1;
hence ( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v ) ; :: thesis: ( (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p )
( (v + w) * p = {} & (v * p) + (w * p) = {} & v * (p + q) = {} & (v * p) + (v * q) = {} & v * (q * p) = {} & (v * q) * p = {} & v * (1_ R2) = {} & v = {} & x * (v * p) = {} & (x * v) * p = {} ) by CARD_1:87, TARSKI:def 1;
hence ( (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p ) ; :: thesis: verum