let X be ComplexUnitarySpace; :: thesis: for seq1, seq2, seq3 being sequence of X holds (seq1 + seq2) - seq3 = seq1 + (seq2 - seq3)
let seq1, seq2, seq3 be sequence of X; :: thesis: (seq1 + seq2) - seq3 = seq1 + (seq2 - seq3)

now :: thesis: for n being Element of NAT holds ((seq1 + seq2) - seq3) . n = (seq1 + (seq2 - seq3)) . n

let n be Element of NAT ; :: thesis: ((seq1 + seq2) - seq3) . n = (seq1 + (seq2 - seq3)) . n
thus ((seq1 + seq2) - seq3) . n = ((seq1 + seq2) . n) - (seq3 . n) by NORMSP_1:def 3
.= ((seq1 . n) + (seq2 . n)) - (seq3 . n) by NORMSP_1:def 2
.= (seq1 . n) + ((seq2 . n) - (seq3 . n)) by RLVECT_1:def 3
.= (seq1 . n) + ((seq2 - seq3) . n) by NORMSP_1:def 3
.= (seq1 + (seq2 - seq3)) . n by NORMSP_1:def 2 ; :: thesis: verum

end;

hence (seq1 + seq2) - seq3 = seq1 + (seq2 - seq3) by FUNCT_2:63; :: thesis: verum