let V be RealLinearSpace; :: thesis: for L1, L2 being Linear_Combination of V holds (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = L1 + L2

let L1, L2 be Linear_Combination of V; :: thesis: (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = L1 + L2

set v2 = vector ((LC_RLSpace V),L2);

A1: ( L1 = @ (@ L1) & L2 = @ (@ L2) ) ;

L2 in the carrier of (LC_RLSpace V) by Def14;

then A2: L2 in LC_RLSpace V ;

L1 in the carrier of (LC_RLSpace V) by Def14;

then L1 in LC_RLSpace V ;

hence (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = (LCAdd V) . [L1,(vector ((LC_RLSpace V),L2))] by Def1

.= (LCAdd V) . ((@ L1),(@ L2)) by A2, Def1

.= L1 + L2 by A1, Def17 ;

:: thesis: verum

let L1, L2 be Linear_Combination of V; :: thesis: (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = L1 + L2

set v2 = vector ((LC_RLSpace V),L2);

A1: ( L1 = @ (@ L1) & L2 = @ (@ L2) ) ;

L2 in the carrier of (LC_RLSpace V) by Def14;

then A2: L2 in LC_RLSpace V ;

L1 in the carrier of (LC_RLSpace V) by Def14;

then L1 in LC_RLSpace V ;

hence (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = (LCAdd V) . [L1,(vector ((LC_RLSpace V),L2))] by Def1

.= (LCAdd V) . ((@ L1),(@ L2)) by A2, Def1

.= L1 + L2 by A1, Def17 ;

:: thesis: verum