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 . n) + ((seq2 + seq3) . n) by NORMSP_1:def 2
.= (seq1 . n) + ((seq2 . n) + (seq3 . n)) by NORMSP_1:def 2
.= ((seq1 . n) + (seq2 . n)) + (seq3 . n) by RLVECT_1:def 3
.= ((seq1 + seq2) . n) + (seq3 . n) by NORMSP_1:def 2
.= ((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