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
A11: dom f1 = dom fi and
A12: for A being Ordinal st A in dom fi holds
f1 . A = C *^ (fi . A) and
A13: dom f2 = dom fi and
A14: for A being Ordinal st A in dom fi holds
f2 . A = C *^ (fi . A) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in dom fi holds
f1 . x = f2 . x

let x be object ; :: thesis: ( x in dom fi implies f1 . x = f2 . x )
assume A15: x in dom fi ; :: thesis: f1 . x = f2 . x
then reconsider A = x as Ordinal ;
thus f1 . x = C *^ (fi . A) by A12, A15
.= f2 . x by A14, A15 ; :: thesis: verum

end;

hence f1 = f2 by A11, A13, FUNCT_1:2; :: thesis: verum