let X be ComplexUnitarySpace; :: thesis: for seq being sequence of X holds seq = seq - (0. X)
let seq be sequence of X; :: thesis: seq = seq - (0. X)

now :: thesis: for n being Element of NAT holds (seq - H₁(X)) . n = seq . n

let n be Element of NAT ; :: thesis: (seq - H₁(X)) . n = seq . n
thus (seq - H₁(X)) . n = (seq . n) - H₁(X) by NORMSP_1:def 4
.= seq . n by RLVECT_1:13 ; :: thesis: verum

end;

hence seq = seq - (0. X) by FUNCT_2:63; :: thesis: verum