let f1, f2 be Ordinal-Sequence; :: thesis: ( dom f1 = dom fi & ( for A being Ordinal st A in dom fi holds
f1 . A = (fi . A) *^ C ) & dom f2 = dom fi & ( for A being Ordinal st A in dom fi holds
f2 . A = (fi . A) *^ C ) implies f1 = f2 )
assume that
A16: dom f1 = dom fi and
A17: for A being Ordinal st A in dom fi holds
f1 . A = (fi . A) *^ C and
A18: dom f2 = dom fi and
A19: for A being Ordinal st A in dom fi holds
f2 . A = (fi . A) *^ C ; :: 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 A20: x in dom fi ; :: thesis: f1 . x = f2 . x
then reconsider A = x as Ordinal ;
thus f1 . x = (fi . A) *^ C by A17, A20
.= f2 . x by A19, A20 ; :: thesis: verum

end;

hence f1 = f2 by A16, A18, FUNCT_1:2; :: thesis: verum