let X be RealLinearSpace; :: thesis: for A, B being Subset of X
for l1 being Linear_Combination of A
for l2 being Linear_Combination of B st rng l1 c= RAT & rng l2 c= RAT & A misses B holds
ex l being Linear_Combination of A \/ B st
( Carrier l = (Carrier l1) \/ (Carrier l2) & rng l c= RAT & Sum l = (Sum l1) + (Sum l2) )
let A, B be Subset of X; :: thesis: for l1 being Linear_Combination of A
for l2 being Linear_Combination of B st rng l1 c= RAT & rng l2 c= RAT & A misses B holds
ex l being Linear_Combination of A \/ B st
( Carrier l = (Carrier l1) \/ (Carrier l2) & rng l c= RAT & Sum l = (Sum l1) + (Sum l2) )
let l1 be Linear_Combination of A; :: thesis: for l2 being Linear_Combination of B st rng l1 c= RAT & rng l2 c= RAT & A misses B holds
ex l being Linear_Combination of A \/ B st
( Carrier l = (Carrier l1) \/ (Carrier l2) & rng l c= RAT & Sum l = (Sum l1) + (Sum l2) )
let l2 be Linear_Combination of B; :: thesis: ( rng l1 c= RAT & rng l2 c= RAT & A misses B implies ex l being Linear_Combination of A \/ B st
( Carrier l = (Carrier l1) \/ (Carrier l2) & rng l c= RAT & Sum l = (Sum l1) + (Sum l2) ) )
assume that
A1: ( rng l1 c= RAT & rng l2 c= RAT ) and
A2: A misses B ; :: thesis: ex l being Linear_Combination of A \/ B st
( Carrier l = (Carrier l1) \/ (Carrier l2) & rng l c= RAT & Sum l = (Sum l1) + (Sum l2) )
consider l being Linear_Combination of A \/ B such that
A3: ( Carrier l = (Carrier l1) \/ (Carrier l2) & l = l1 + l2 ) by A2, Th7;
take l ; :: thesis: ( Carrier l = (Carrier l1) \/ (Carrier l2) & rng l c= RAT & Sum l = (Sum l1) + (Sum l2) )
thus Carrier l = (Carrier l1) \/ (Carrier l2) by A3; :: thesis: ( rng l c= RAT & Sum l = (Sum l1) + (Sum l2) )

now :: thesis: for y being object st y in rng l holds
y in RAT

let y be object ; :: thesis: ( y in rng l implies y in RAT )
assume y in rng l ; :: thesis: y in RAT
then consider x being object such that
A4: ( x in dom l & y = l . x ) by FUNCT_1:def 3;
A5: x in the carrier of X by A4;
A6: l . x = (l1 . x) + (l2 . x) by A3, A4, RLVECT_2:def 10;
( x in dom l1 & x in dom l2 ) by A5, FUNCT_2:def 1;
then ( l1 . x in rng l1 & l2 . x in rng l2 ) by FUNCT_1:3;
hence y in RAT by A1, A4, A6, RAT_1:def 2; :: thesis: verum

end;

hence rng l c= RAT ; :: thesis: Sum l = (Sum l1) + (Sum l2)
thus Sum l = (Sum l1) + (Sum l2) by A3, RLVECT_3:1; :: thesis: verum