let seq1, seq2 be Real_Sequence; :: thesis: ( seq1 = - seq2 iff for n being Nat holds seq1 . n = - (seq2 . n) )

thus ( seq1 = - seq2 implies for n being Nat holds seq1 . n = - (seq2 . n) ) by VALUED_1:8; :: thesis: ( ( for n being Nat holds seq1 . n = - (seq2 . n) ) implies seq1 = - seq2 )

assume for n being Nat holds seq1 . n = - (seq2 . n) ; :: thesis: seq1 = - seq2

then A1: for n being object st n in dom seq1 holds

seq1 . n = - (seq2 . n) ;

dom seq1 = NAT by FUNCT_2:def 1

.= dom seq2 by FUNCT_2:def 1 ;

hence seq1 = - seq2 by A1, VALUED_1:9; :: thesis: verum

thus ( seq1 = - seq2 implies for n being Nat holds seq1 . n = - (seq2 . n) ) by VALUED_1:8; :: thesis: ( ( for n being Nat holds seq1 . n = - (seq2 . n) ) implies seq1 = - seq2 )

assume for n being Nat holds seq1 . n = - (seq2 . n) ; :: thesis: seq1 = - seq2

then A1: for n being object st n in dom seq1 holds

seq1 . n = - (seq2 . n) ;

dom seq1 = NAT by FUNCT_2:def 1

.= dom seq2 by FUNCT_2:def 1 ;

hence seq1 = - seq2 by A1, VALUED_1:9; :: thesis: verum