let K be non empty addLoopStr ; :: thesis: for V, W being non empty VectSpStr of K
for v, u being Vector of V
for w being Vector of W
for f being Form of V,W st f is additiveSAF holds
f . ((v + u),w) = (f . (v,w)) + (f . (u,w))
let V, W be non empty VectSpStr of K; :: thesis: for v, u being Vector of V
for w being Vector of W
for f being Form of V,W st f is additiveSAF holds
f . ((v + u),w) = (f . (v,w)) + (f . (u,w))
let v, w be Vector of V; :: thesis: for w being Vector of W
for f being Form of V,W st f is additiveSAF holds
f . ((v + w),w) = (f . (v,w)) + (f . (w,w))
let y be Vector of W; :: thesis: for f being Form of V,W st f is additiveSAF holds
f . ((v + w),y) = (f . (v,y)) + (f . (w,y))
let f be Form of V,W; :: thesis: ( f is additiveSAF implies f . ((v + w),y) = (f . (v,y)) + (f . (w,y)) )
set F = FunctionalSAF (f,y);
assume f is additiveSAF ; :: thesis: f . ((v + w),y) = (f . (v,y)) + (f . (w,y))
then A1: FunctionalSAF (f,y) is additive by Def13;
thus f . ((v + w),y) = (FunctionalSAF (f,y)) . (v + w) by Th10
.= ((FunctionalSAF (f,y)) . v) + ((FunctionalSAF (f,y)) . w) by A1, GRCAT_1:def 8
.= (f . (v,y)) + ((FunctionalSAF (f,y)) . w) by Th10
.= (f . (v,y)) + (f . (w,y)) by Th10 ; :: thesis: verum