defpred S₁[ object , object ] means ex R being TolStr st
( R = C . $1 & $2 = the ToleranceRel of R );
A1: for i being object st i in I holds
ex j being object st S₁[i,j]

proof

let i be object ; :: thesis: ( i in I implies ex j being object st S₁[i,j] )
assume A2: i in I ; :: thesis: ex j being object st S₁[i,j]
then reconsider I = I as non empty set ;
reconsider B = C as () ManySortedSet of I ;
reconsider i9 = i as Element of I by A2;
take the ToleranceRel of (B . i9) ; :: thesis: S₁[i, the ToleranceRel of (B . i9)]
take B . i9 ; :: thesis: ( B . i9 = C . i & the ToleranceRel of (B . i9) = the ToleranceRel of (B . i9) )
thus ( B . i9 = C . i & the ToleranceRel of (B . i9) = the ToleranceRel of (B . i9) ) ; :: thesis: verum

end;

consider M being Function such that
A3: dom M = I and
A4: for i being object st i in I holds
S₁[i,M . i] from CLASSES1:sch 1(A1);
reconsider M = M as ManySortedSet of I by A3, PARTFUN1:def 2, RELAT_1:def 18;
take M ; :: thesis: for i being set st i in I holds
ex P being TolStr st
( P = C . i & M . i = the ToleranceRel of P )
thus for i being set st i in I holds
ex P being TolStr st
( P = C . i & M . i = the ToleranceRel of P ) by A4; :: thesis: verum