let V be RealLinearSpace; :: thesis: for v1, v2 being VECTOR of V st v1 <> v2 holds
for l being Linear_Combination of {v1,v2} holds Sum l = ((l . v1) * v1) + ((l . v2) * v2)
let v1, v2 be VECTOR of V; :: thesis: ( v1 <> v2 implies for l being Linear_Combination of {v1,v2} holds Sum l = ((l . v1) * v1) + ((l . v2) * v2) )
assume A1: v1 <> v2 ; :: thesis: for l being Linear_Combination of {v1,v2} holds Sum l = ((l . v1) * v1) + ((l . v2) * v2)
let l be Linear_Combination of {v1,v2}; :: thesis: Sum l = ((l . v1) * v1) + ((l . v2) * v2)
A2: Carrier l c= {v1,v2} by Def8;

now

per cases ( Carrier l = {} or Carrier l = {v1} or Carrier l = {v2} or Carrier l = {v1,v2} ) by A2, ZFMISC_1:42;

suppose Carrier l = {} ; :: thesis: Sum l = ((l . v1) * v1) + ((l . v2) * v2)

then A3: l = ZeroLC V by Def7;
hence Sum l = 0. V by Lm1
.= (0. V) + (0. V) by RLVECT_1:10
.= (0 * v1) + (0. V) by RLVECT_1:23
.= (0 * v1) + (0 * v2) by RLVECT_1:23
.= ((l . v1) * v1) + (0 * v2) by A3, Th30
.= ((l . v1) * v1) + ((l . v2) * v2) by A3, Th30 ;
:: thesis: verum

end;

suppose A4: Carrier l = {v1} ; :: thesis: Sum l = ((l . v1) * v1) + ((l . v2) * v2)

then reconsider L = l as Linear_Combination of {v1} by Def8;
A5: not v2 in Carrier l by A1, A4, TARSKI:def 1;
thus Sum l = Sum L
.= (l . v1) * v1 by Th50
.= ((l . v1) * v1) + (0. V) by RLVECT_1:10
.= ((l . v1) * v1) + (0 * v2) by RLVECT_1:23
.= ((l . v1) * v1) + ((l . v2) * v2) by A5 ; :: thesis: verum

end;

suppose A6: Carrier l = {v2} ; :: thesis: Sum l = ((l . v1) * v1) + ((l . v2) * v2)

then reconsider L = l as Linear_Combination of {v2} by Def8;
A7: not v1 in Carrier l by A1, A6, TARSKI:def 1;
thus Sum l = Sum L
.= (l . v2) * v2 by Th50
.= (0. V) + ((l . v2) * v2) by RLVECT_1:10
.= (0 * v1) + ((l . v2) * v2) by RLVECT_1:23
.= ((l . v1) * v1) + ((l . v2) * v2) by A7 ; :: thesis: verum

end;

suppose Carrier l = {v1,v2} ; :: thesis: Sum l = ((l . v1) * v1) + ((l . v2) * v2)

then consider F being FinSequence of the carrier of V such that
A8: ( F is one-to-one & rng F = {v1,v2} ) and
A9: Sum l = Sum (l (#) F) by Def10;
( F = <*v1,v2*> or F = <*v2,v1*> ) by A1, A8, FINSEQ_3:108;
then ( l (#) F = <*((l . v1) * v1),((l . v2) * v2)*> or l (#) F = <*((l . v2) * v2),((l . v1) * v1)*> ) by Th43;
hence Sum l = ((l . v1) * v1) + ((l . v2) * v2) by A9, RLVECT_1:62; :: thesis: verum

end;

end;

end;

hence Sum l = ((l . v1) * v1) + ((l . v2) * v2) ; :: thesis: verum