defpred S1[ object , object ] means ex n being Nat st
( n = \$1 & \$2 = F1(n) );
A1: now :: thesis: for x being object st x in NAT holds
ex y being object st S1[x,y]
let x be object ; :: thesis: ( x in NAT implies ex y being object st S1[x,y] )
assume x in NAT ; :: thesis: ex y being object st S1[x,y]
then consider n being Nat such that
A2: n = x ;
reconsider r2 = F1(n) as object ;
take y = r2; :: thesis: S1[x,y]
thus S1[x,y] by A2; :: thesis: verum
end;
consider f being Function such that
A3: dom f = NAT and
A4: for x being object st x in NAT holds
S1[x,f . x] from
now :: thesis: for x being object st x in NAT holds
f . x is real
let x be object ; :: thesis: ( x in NAT implies f . x is real )
assume x in NAT ; :: thesis: f . x is real
then ex n being Nat st
( n = x & f . x = F1(n) ) by A4;
hence f . x is real ; :: thesis: verum
end;
then reconsider f = f as Real_Sequence by ;
take seq = f; :: thesis: for n being Nat holds seq . n = F1(n)
let n be Nat; :: thesis: seq . n = F1(n)
n in NAT by ORDINAL1:def 12;
then ex k being Nat st
( k = n & seq . n = F1(k) ) by A4;
hence seq . n = F1(n) ; :: thesis: verum