let seq1, seq2, seq3 be Real_Sequence; :: 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 Th7
.= ((seq1 . n) + (seq2 . n)) + (seq3 . n) by Th7
.= (seq1 . n) + ((seq2 . n) + (seq3 . n))
.= (seq1 . n) + ((seq2 + seq3) . n) by Th7
.= (seq1 + (seq2 + seq3)) . n by Th7 ; :: thesis: verum

end;

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