set p = <*1*>;
{1} c= INT by Lm7, TARSKI:def 3;
then rng <*1*> c= INT by FINSEQ_1:39;
then reconsider f = <*1*> as FinSequence of INT by FINSEQ_1:def 4;
take f ; :: thesis: ( not f is empty & f is positive-yielding & f is Chinese_Remainder )
A1: now :: thesis: for i being Element of NAT st i in dom f holds
i = 1
let i be Element of NAT ; :: thesis: ( i in dom f implies i = 1 )
assume i in dom f ; :: thesis: i = 1
then i in Seg 1 by FINSEQ_1:38;
hence i = 1 by FINSEQ_1:2, TARSKI:def 1; :: thesis: verum
end;
A2: now :: thesis: for i9, j9 being Nat st i9 in dom f & j9 in dom f & i9 <> j9 holds
f . i9,f . j9 are_coprime
let i9, j9 be Nat; :: thesis: ( i9 in dom f & j9 in dom f & i9 <> j9 implies f . i9,f . j9 are_coprime )
assume that
A3: i9 in dom f and
A4: j9 in dom f and
A5: i9 <> j9 ; :: thesis: f . i9,f . j9 are_coprime
reconsider i = i9, j = j9 as Element of NAT by ORDINAL1:def 12;
i = 1 by A1, A3
.= j by A1, A4 ;
hence f . i9,f . j9 are_coprime by A5; :: thesis: verum
end;
now :: thesis: for r being Real st r in rng f holds
0 < r
let r be Real; :: thesis: ( r in rng f implies 0 < r )
assume r in rng f ; :: thesis: 0 < r
then r in {1} by FINSEQ_1:39;
hence 0 < r by TARSKI:def 1; :: thesis: verum
end;
hence ( not f is empty & f is positive-yielding & f is Chinese_Remainder ) by A2, PARTFUN3:def 1; :: thesis: verum