consider seq being Real_Sequence such that
A1: for n being Nat holds seq . n = F₁(n) from SEQ_1:sch 1();
thus ex seq being Real_Sequence st
for n being Nat holds seq . n = F₁(n) by A1; :: thesis: for seq1, seq2 being Real_Sequence st ( for n being Nat holds seq1 . n = F₁(n) ) & ( for n being Nat holds seq2 . n = F₁(n) ) holds
seq1 = seq2
let seq1, seq2 be Real_Sequence; :: thesis: ( ( for n being Nat holds seq1 . n = F₁(n) ) & ( for n being Nat holds seq2 . n = F₁(n) ) implies seq1 = seq2 )
assume that
A2: for n being Nat holds seq1 . n = F₁(n) and
A3: for n being Nat holds seq2 . n = F₁(n) ; :: thesis: seq1 = seq2
let n be Element of NAT ; :: according to FUNCT_2:def 8 :: thesis: seq1 . n = seq2 . n
thus seq1 . n = F₁(n) by A2
.= seq2 . n by A3 ; :: thesis: verum