set qa = QuotOSAlg (U1,R);
set cqa = the Sorts of (QuotOSAlg (U1,R));
set S1 = the Sorts of U1;
set S2 = the Sorts of U2;
defpred S₁[ object , object ] means for a being Element of the Sorts of U1 . s st $1 = OSClass (R,a) holds
$2 = (F . s) . a;
A4: the Sorts of (QuotOSAlg (U1,R)) . s = OSClass (R,s) by Def11;
A5: for x being object st x in the Sorts of (QuotOSAlg (U1,R)) . s holds
ex y being object st
( y in the Sorts of U2 . s & S₁[x,y] )

proof

let x be object ; :: thesis: ( x in the Sorts of (QuotOSAlg (U1,R)) . s implies ex y being object st
( y in the Sorts of U2 . s & S₁[x,y] ) )
A6: R . s c= (OSCng F) . s by A3, PBOOLE:def 2;
assume x in the Sorts of (QuotOSAlg (U1,R)) . s ; :: thesis: ex y being object st
( y in the Sorts of U2 . s & S₁[x,y] )
then consider a being set such that
A7: a in the Sorts of U1 . s and
A8: x = Class ((CompClass (R,(CComp s))),a) by A4, Def10;
reconsider a = a as Element of the Sorts of U1 . s by A7;
take y = (F . s) . a; :: thesis: ( y in the Sorts of U2 . s & S₁[x,y] )
thus y in the Sorts of U2 . s ; :: thesis: S₁[x,y]
let b be Element of the Sorts of U1 . s; :: thesis: ( x = OSClass (R,b) implies y = (F . s) . b )
assume A9: x = OSClass (R,b) ; :: thesis: y = (F . s) . b
x = OSClass (R,a) by A8;
then [b,a] in R . s by A9, Th12;
then [b,a] in (OSCng F) . s by A6;
then [b,a] in (MSCng F) . s by A1, A2, Def23;
then [b,a] in MSCng (F,s) by A1, MSUALG_4:def 18;
hence y = (F . s) . b by MSUALG_4:def 17; :: thesis: verum

end;

consider G being Function such that
A10: ( dom G = the Sorts of (QuotOSAlg (U1,R)) . s & rng G c= the Sorts of U2 . s & ( for x being object st x in the Sorts of (QuotOSAlg (U1,R)) . s holds
S₁[x,G . x] ) ) from FUNCT_1:sch 6(A5);
reconsider G = G as Function of ( the Sorts of (QuotOSAlg (U1,R)) . s),( the Sorts of U2 . s) by A10, FUNCT_2:def 1, RELSET_1:4;
take G ; :: thesis: for x being Element of the Sorts of U1 . s holds G . (OSClass (R,x)) = (F . s) . x
let a be Element of the Sorts of U1 . s; :: thesis: G . (OSClass (R,a)) = (F . s) . a
thus G . (OSClass (R,a)) = (F . s) . a by A4, A10; :: thesis: verum