defpred S₁[ set , set ] means $2 = u mod (m . $1);
A1: for k being Nat st k in Seg (len m) holds
ex x being Element of INT st S₁[k,x]

proof

let k be Nat; :: thesis: ( k in Seg (len m) implies ex x being Element of INT st S₁[k,x] )
assume k in Seg (len m) ; :: thesis: ex x being Element of INT st S₁[k,x]
reconsider j = u mod (m . k) as Element of INT by INT_1:def 2;
take j ; :: thesis: S₁[k,j]
thus S₁[k,j] ; :: thesis: verum

end;

consider p being FinSequence of INT such that
A2: ( dom p = Seg (len m) & ( for k being Nat st k in Seg (len m) holds
S₁[k,p . k] ) ) from FINSEQ_1:sch 5(A1);
take p ; :: thesis: ( len p = len m & ( for i being Nat st i in dom p holds
p . i = u mod (m . i) ) )
thus len p = len m by A2, FINSEQ_1:def 3; :: thesis: for i being Nat st i in dom p holds
p . i = u mod (m . i)
thus for i being Nat st i in dom p holds
p . i = u mod (m . i) by A2; :: thesis: verum