set O = OwnSymbolsOf S;
set SS = AllSymbolsOf S;
let IT1, IT2 be Function; :: thesis: ( dom IT1 = OwnSymbolsOf S & ( for s being own Element of S holds IT1 . s = X -freeInterpreter s ) & dom IT2 = OwnSymbolsOf S & ( for s being own Element of S holds IT2 . s = X -freeInterpreter s ) implies IT1 = IT2 )
assume A3: ( dom IT1 = OwnSymbolsOf S & ( for s being own Element of S holds IT1 . s = X -freeInterpreter s ) ) ; :: thesis: ( not dom IT2 = OwnSymbolsOf S or ex s being own Element of S st not IT2 . s = X -freeInterpreter s or IT1 = IT2 )
assume A4: ( dom IT2 = OwnSymbolsOf S & ( for s being own Element of S holds IT2 . s = X -freeInterpreter s ) ) ; :: thesis: IT1 = IT2

now :: thesis: ( dom IT1 = dom IT2 & ( for x being set st x in dom IT1 holds
IT1 . x = IT2 . x ) )

thus dom IT1 = dom IT2 by A3, A4; :: thesis: for x being set st x in dom IT1 holds
IT1 . x = IT2 . x
let x be set ; :: thesis: ( x in dom IT1 implies IT1 . x = IT2 . x )
assume x in dom IT1 ; :: thesis: IT1 . x = IT2 . x
then reconsider s = x as own Element of S by A3, FOMODEL1:def 19;
IT1 . s = X -freeInterpreter s by A3
.= IT2 . s by A4 ;
hence IT1 . x = IT2 . x ; :: thesis: verum

end;

hence IT1 = IT2 by FUNCT_1:2; :: thesis: verum