let X be RealUnitarySpace; :: thesis: for seq1, seq2 being sequence of X holds seq1 - seq2 = seq1 + (- seq2)
let seq1, seq2 be sequence of X; :: thesis: seq1 - seq2 = seq1 + (- seq2)

now

let n be Element of NAT ; :: thesis: (seq1 - seq2) . n = (seq1 + (- seq2)) . n
thus (seq1 - seq2) . n = (seq1 . n) - (seq2 . n) by NORMSP_1:def 6
.= (seq1 . n) + ((- seq2) . n) by Def10
.= (seq1 + (- seq2)) . n by NORMSP_1:def 5 ; :: thesis: verum

end;

hence seq1 - seq2 = seq1 + (- seq2) by FUNCT_2:113; :: thesis: verum