set p = <*1*>;

now

let u be set ; :: thesis: ( u in {1} implies u in INT )
assume u in {1} ; :: thesis: u in INT
then reconsider u' = u as Element of NAT by TARSKI:def 1;
u' is integer number ;
hence u in INT by INT_1:def 1; :: thesis: verum

end;

then {1} c= INT by TARSKI:def 3;
then rng <*1*> c= INT by FINSEQ_1:56;
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 )

H: now

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_1:55;
hence i = 1 by FINSEQ_1:4, TARSKI:def 1; :: thesis: verum

end;

C: now

let r be real number ; :: thesis: ( r in rng f implies 0 < r )
assume r in rng f ; :: thesis: 0 < r
then r in {1} by FINSEQ_1:56;
hence 0 < r by TARSKI:def 1; :: thesis: verum

end;

now

let i', j' be natural number ; :: thesis: ( i' in dom f & j' in dom f & i' <> j' implies f . i',f . j' are_relative_prime )
assume D1: ( i' in dom f & j' in dom f & i' <> j' ) ; :: thesis: f . i',f . j' are_relative_prime
reconsider i = i', j = j' as Element of NAT by ORDINAL1:def 13;
i = 1 by D1, H
.= j by D1, H ;
hence f . i',f . j' are_relative_prime by D1; :: thesis: verum

end;

hence ( not f is empty & f is positive-yielding & f is Chinese_Remainder ) by C, DefCR, PARTFUN3:def 1; :: thesis: verum