set qa = QuotOSAlg U1,R;
set cqa = the Sorts of (QuotOSAlg U1,R);
set u1 = the Sorts of U1;
set u2 = the Sorts of U2;
A8: the Sorts of (QuotOSAlg U1,R) . s = OSClass R,s by Def13;
let H, G be Function of (the Sorts of (QuotOSAlg U1,R) . s),(the Sorts of U2 . s); :: thesis: ( ( for x being Element of the Sorts of U1 . s holds H . (OSClass R,x) = (F . s) . x ) & ( for x being Element of the Sorts of U1 . s holds G . (OSClass R,x) = (F . s) . x ) implies H = G )
assume that
A9: for a being Element of the Sorts of U1 . s holds H . (OSClass R,a) = (F . s) . a and
A10: for a being Element of the Sorts of U1 . s holds G . (OSClass R,a) = (F . s) . a ; :: thesis: H = G
for x being set st x in the Sorts of (QuotOSAlg U1,R) . s holds
H . x = G . x

proof

let x be set ; :: thesis: ( x in the Sorts of (QuotOSAlg U1,R) . s implies H . x = G . x )
assume x in the Sorts of (QuotOSAlg U1,R) . s ; :: thesis: H . x = G . x
then consider y being set such that
A11: ( y in the Sorts of U1 . s & x = Class (CompClass R,(CComp s)),y ) by A8, Def12;
reconsider y1 = y as Element of the Sorts of U1 . s by A11;
A12: OSClass R,y1 = x by A11;
hence H . x = (F . s) . y1 by A9
.= G . x by A10, A12 ;
:: thesis: verum

end;

hence H = G by FUNCT_2:18; :: thesis: verum