let R be Ring; :: thesis: for V being RightMod of R
for S, T being finite Subset of V holds Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))
let V be RightMod of R; :: thesis: for S, T being finite Subset of V holds Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))
let S, T be finite Subset of V; :: thesis: Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))
set A = S \ T;
set B = T \ S;
set Z = (S \ T) \/ (T \ S);
set I = T /\ S;
A1: (S \ T) \/ (T /\ S) = S by XBOOLE_1:51;
A2: (T \ S) \/ (T /\ S) = T by XBOOLE_1:51;
A3: (S \ T) \/ (T \ S) = T \+\ S ;
then ((S \ T) \/ (T \ S)) \/ (T /\ S) = T \/ S by XBOOLE_1:93;
then (Sum (T \/ S)) + (Sum (T /\ S)) = ((Sum ((S \ T) \/ (T \ S))) + (Sum (T /\ S))) + (Sum (T /\ S)) by A3, Th9, XBOOLE_1:103
.= (((Sum (S \ T)) + (Sum (T \ S))) + (Sum (T /\ S))) + (Sum (T /\ S)) by Th9, XBOOLE_1:82
.= ((Sum (S \ T)) + ((Sum (T /\ S)) + (Sum (T \ S)))) + (Sum (T /\ S)) by RLVECT_1:def 3
.= ((Sum (S \ T)) + (Sum (T /\ S))) + ((Sum (T \ S)) + (Sum (T /\ S))) by RLVECT_1:def 3
.= (Sum S) + ((Sum (T \ S)) + (Sum (T /\ S))) by A1, Th9, XBOOLE_1:89
.= (Sum T) + (Sum S) by A2, Th9, XBOOLE_1:89 ;
hence Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S)) by RLSUB_2:61; :: thesis: verum