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
A4: ( dom f1 = dom fi & ( for A being Ordinal st A in dom fi holds
f1 . A = (fi . A) +^ C ) ) and
A5: ( dom f2 = dom fi & ( for A being Ordinal st A in dom fi holds
f2 . A = (fi . A) +^ C ) ) ; :: thesis: f1 = f2

now

let x be set ; :: thesis: ( x in dom fi implies f1 . x = f2 . x )
assume A6: x in dom fi ; :: thesis: f1 . x = f2 . x
then reconsider A = x as Ordinal ;
thus f1 . x = (fi . A) +^ C by A4, A6
.= f2 . x by A5, A6 ; :: thesis: verum

end;

hence f1 = f2 by A4, A5, FUNCT_1:9; :: thesis: verum