let X be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; :: thesis: for seq1, seq2 being sequence of X holds (Partial_Sums seq1) - (Partial_Sums seq2) = Partial_Sums (seq1 - seq2)
let seq1, seq2 be sequence of X; :: thesis: (Partial_Sums seq1) - (Partial_Sums seq2) = Partial_Sums (seq1 - seq2)
set PSseq1 = Partial_Sums seq1;
set PSseq2 = Partial_Sums seq2;

A1: now :: thesis: for n being Nat holds ((Partial_Sums seq1) - (Partial_Sums seq2)) . (n + 1) = (((Partial_Sums seq1) - (Partial_Sums seq2)) . n) + ((seq1 - seq2) . (n + 1))

let n be Nat; :: thesis: ((Partial_Sums seq1) - (Partial_Sums seq2)) . (n + 1) = (((Partial_Sums seq1) - (Partial_Sums seq2)) . n) + ((seq1 - seq2) . (n + 1))
thus ((Partial_Sums seq1) - (Partial_Sums seq2)) . (n + 1) = ((Partial_Sums seq1) . (n + 1)) - ((Partial_Sums seq2) . (n + 1)) by NORMSP_1:def 3
.= (((Partial_Sums seq1) . n) + (seq1 . (n + 1))) - ((Partial_Sums seq2) . (n + 1)) by Def1
.= (((Partial_Sums seq1) . n) + (seq1 . (n + 1))) - ((seq2 . (n + 1)) + ((Partial_Sums seq2) . n)) by Def1
.= ((((Partial_Sums seq1) . n) + (seq1 . (n + 1))) - (seq2 . (n + 1))) - ((Partial_Sums seq2) . n) by RLVECT_1:27
.= (((Partial_Sums seq1) . n) + ((seq1 . (n + 1)) - (seq2 . (n + 1)))) - ((Partial_Sums seq2) . n) by RLVECT_1:def 3
.= (((Partial_Sums seq1) . n) - ((Partial_Sums seq2) . n)) + ((seq1 . (n + 1)) - (seq2 . (n + 1))) by RLVECT_1:def 3
.= (((Partial_Sums seq1) - (Partial_Sums seq2)) . n) + ((seq1 . (n + 1)) - (seq2 . (n + 1))) by NORMSP_1:def 3
.= (((Partial_Sums seq1) - (Partial_Sums seq2)) . n) + ((seq1 - seq2) . (n + 1)) by NORMSP_1:def 3 ; :: thesis: verum

end;

((Partial_Sums seq1) - (Partial_Sums seq2)) . 0 = ((Partial_Sums seq1) . 0) - ((Partial_Sums seq2) . 0) by NORMSP_1:def 3
.= (seq1 . 0) - ((Partial_Sums seq2) . 0) by Def1
.= (seq1 . 0) - (seq2 . 0) by Def1
.= (seq1 - seq2) . 0 by NORMSP_1:def 3 ;
hence (Partial_Sums seq1) - (Partial_Sums seq2) = Partial_Sums (seq1 - seq2) by A1, Def1; :: thesis: verum