begin
Lm1:
{} in omega
by ORDINAL1:def 11;
Lm2:
omega is limit_ordinal
by ORDINAL1:def 11;
Lm3:
1 = succ {}
;
Lm4:
for fi being Ordinal-Sequence
for A being Ordinal st A is_limes_of fi holds
dom fi <> {}
Lm5:
for f, g being Function
for X being set st rng f c= X holds
(g | X) * f = g * f
Lm6:
for A being Ordinal st A <> {} & A is limit_ordinal holds
for fi being Ordinal-Sequence st dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp ({},B) ) holds
{} is_limes_of fi
Lm7:
for A being Ordinal st A <> {} holds
for fi being Ordinal-Sequence st dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp (1,B) ) holds
1 is_limes_of fi
Lm8:
for C, A being Ordinal st A <> {} & A is limit_ordinal holds
ex fi being Ordinal-Sequence st
( dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp (C,B) ) & ex D being Ordinal st D is_limes_of fi )
begin