let f1, f2 be Ordinal-Sequence; :: thesis: ( dom f1 = dom fi & ( for A being Ordinal st A in dom fi holds
f1 . A = C *^ (fi . A) ) & dom f2 = dom fi & ( for A being Ordinal st A in dom fi holds
f2 . A = C *^ (fi . A) ) implies f1 = f2 )
assume that
A7: ( dom f1 = dom fi & ( for A being Ordinal st A in dom fi holds
f1 . A = C *^ (fi . A) ) ) and
A8: ( dom f2 = dom fi & ( for A being Ordinal st A in dom fi holds
f2 . A = C *^ (fi . A) ) ) ; :: thesis: f1 = f2

now

let x be set ; :: thesis: ( x in dom fi implies f1 . x = f2 . x )
assume A9: x in dom fi ; :: thesis: f1 . x = f2 . x
then reconsider A = x as Ordinal ;
thus f1 . x = C *^ (fi . A) by A7, A9
.= f2 . x by A8, A9 ; :: thesis: verum

end;

hence f1 = f2 by A7, A8, FUNCT_1:9; :: thesis: verum