let X be RealUnitarySpace; :: thesis: for seq being sequence of X holds Sum seq,1 = (Sum seq,0 ) + (seq . 1)
let seq be sequence of X; :: thesis: Sum seq,1 = (Sum seq,0 ) + (seq . 1)
(Partial_Sums seq) . 1 = ((Partial_Sums seq) . 0 ) + (seq . (0 + 1)) by Def1
.= ((Partial_Sums seq) . 0 ) + (seq . 1) ;
hence Sum seq,1 = (Sum seq,0 ) + (seq . 1) ; :: thesis: verum