let X be RealLinearSpace; :: thesis: for A, B being Subset of X
for l being Linear_Combination of A \/ B st rng l c= RAT & A misses B holds
ex l1 being Linear_Combination of A ex l2 being Linear_Combination of B st
( rng l1 c= RAT & rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) )
let A, B be Subset of X; :: thesis: for l being Linear_Combination of A \/ B st rng l c= RAT & A misses B holds
ex l1 being Linear_Combination of A ex l2 being Linear_Combination of B st
( rng l1 c= RAT & rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) )
let l be Linear_Combination of A \/ B; :: thesis: ( rng l c= RAT & A misses B implies ex l1 being Linear_Combination of A ex l2 being Linear_Combination of B st
( rng l1 c= RAT & rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) ) )
assume that
A1: rng l c= RAT and
A3: A misses B ; :: thesis: ex l1 being Linear_Combination of A ex l2 being Linear_Combination of B st
( rng l1 c= RAT & rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) )
consider l1 being Linear_Combination of A, l2 being Linear_Combination of B such that
A4: ( Carrier l = (Carrier l1) \/ (Carrier l2) & l = l1 + l2 & Carrier l1 = (Carrier l) \ B & Carrier l2 = (Carrier l) \ A ) by A3, Th9;
take l1 ; :: thesis: ex l2 being Linear_Combination of B st
( rng l1 c= RAT & rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) )
take l2 ; :: thesis: ( rng l1 c= RAT & rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) )
A5: ( Carrier l1 c= A & Carrier l2 c= B ) by RLVECT_2:def 6;

now :: thesis: for y being object st y in rng l1 holds
y in RAT

let y be object ; :: thesis: ( y in rng l1 implies b₁ in RAT )
assume y in rng l1 ; :: thesis: b₁ in RAT
then consider x being object such that
A6: ( x in dom l1 & y = l1 . x ) by FUNCT_1:def 3;
x in the carrier of X by A6;
then ( x in dom l & x in dom l2 ) by FUNCT_2:def 1;
then A7: ( l . x in rng l & l2 . x in rng l2 & l . x = (l1 . x) + (l2 . x) ) by A4, FUNCT_1:3, RLVECT_2:def 10;

per cases ( x in Carrier l1 or not x in Carrier l1 ) ;

suppose x in Carrier l1 ; :: thesis: b₁ in RAT

then not x in Carrier l2 by A3, A5, XBOOLE_0:3;
then l2 . x = 0 by A6;
hence y in RAT by A1, A6, A7; :: thesis: verum

end;

suppose not x in Carrier l1 ; :: thesis: b₁ in RAT

then l1 . x = 0 by A6;
hence y in RAT by A6, RAT_1:def 2; :: thesis: verum

end;

end;

end;

hence rng l1 c= RAT ; :: thesis: ( rng l2 c= RAT & Sum l = (Sum l1) + (Sum l2) )

now :: thesis: for y being object st y in rng l2 holds
y in RAT

let y be object ; :: thesis: ( y in rng l2 implies b₁ in RAT )
assume y in rng l2 ; :: thesis: b₁ in RAT
then consider x being object such that
A8: ( x in dom l2 & y = l2 . x ) by FUNCT_1:def 3;
x in the carrier of X by A8;
then ( x in dom l & x in dom l1 ) by FUNCT_2:def 1;
then A9: ( l . x in rng l & l1 . x in rng l1 & l . x = (l1 . x) + (l2 . x) ) by A4, FUNCT_1:3, RLVECT_2:def 10;

per cases ( x in Carrier l2 or not x in Carrier l2 ) ;

suppose x in Carrier l2 ; :: thesis: b₁ in RAT

then not x in Carrier l1 by A3, A5, XBOOLE_0:3;
then l1 . x = 0 by A8;
hence y in RAT by A1, A8, A9; :: thesis: verum

end;

suppose not x in Carrier l2 ; :: thesis: b₁ in RAT

then l2 . x = 0 by A8;
hence y in RAT by A8, RAT_1:def 2; :: thesis: verum

end;

end;

end;

hence rng l2 c= RAT ; :: thesis: Sum l = (Sum l1) + (Sum l2)
thus Sum l = (Sum l1) + (Sum l2) by A4, RLVECT_3:1; :: thesis: verum