let R be Ring; :: thesis: for F being strict LModMorphism of R ex G, H being LeftMod of R ex f being Function of G,H st
( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )
let F be strict LModMorphism of R; :: thesis: ex G, H being LeftMod of R ex f being Function of G,H st
( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )
consider G, H being LeftMod of R such that
A1: F is Morphism of G,H by Th9;
reconsider F9 = F as Morphism of G,H by A1;
consider f being Function of G,H such that
A2: ( LModMorphismStr(# the Dom of F9, the Cod of F9, the Fun of F9 #) = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous ) by Th7;
take G ; :: thesis: ex H being LeftMod of R ex f being Function of G,H st
( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )
take H ; :: thesis: ex f being Function of G,H st
( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )
take f ; :: thesis: ( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )
thus ( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous ) by A2; :: thesis: verum