let fi, psi be Ordinal-Sequence; :: thesis: for A being Ordinal st A is_limes_of psi holds
A is_limes_of fi ^ psi
let A be Ordinal; :: thesis: ( A is_limes_of psi implies A is_limes_of fi ^ psi )
assume A1: ( ( A = 0 & ex B being Ordinal st
( B in dom psi & ( for C being Ordinal st B c= C & C in dom psi holds
psi . C = 0 ) ) ) or ( A <> 0 & ( for B, C being Ordinal st B in A & A in C holds
ex D being Ordinal st
( D in dom psi & ( for E being Ordinal st D c= E & E in dom psi holds
( B in psi . E & psi . E in C ) ) ) ) ) ) ; :: according to ORDINAL2:def 9 :: thesis: A is_limes_of fi ^ psi
A2: dom (fi ^ psi) = (dom fi) +^ (dom psi) by Def1;

per cases ( A = 0 or A <> 0 ) ;
:: according to ORDINAL2:def 9

case A = 0 ; :: thesis: ex b₁ being set st
( b₁ in dom (fi ^ psi) & ( for b₂ being set holds
( not b₁ c= b₂ or not b₂ in dom (fi ^ psi) or (fi ^ psi) . b₂ = 0 ) ) )

then consider B being Ordinal such that
A3: B in dom psi and
A4: for C being Ordinal st B c= C & C in dom psi holds
psi . C = {} by A1;
take B1 = (dom fi) +^ B; :: thesis: ( B1 in dom (fi ^ psi) & ( for b₁ being set holds
( not B1 c= b₁ or not b₁ in dom (fi ^ psi) or (fi ^ psi) . b₁ = 0 ) ) )
thus B1 in dom (fi ^ psi) by A2, A3, ORDINAL2:32; :: thesis: for b₁ being set holds
( not B1 c= b₁ or not b₁ in dom (fi ^ psi) or (fi ^ psi) . b₁ = 0 )
let C be Ordinal; :: thesis: ( not B1 c= C or not C in dom (fi ^ psi) or (fi ^ psi) . C = 0 )
assume that
A5: B1 c= C and
A6: C in dom (fi ^ psi) ; :: thesis: (fi ^ psi) . C = 0
A7: C = B1 +^ (C -^ B1) by A5, ORDINAL3:def 5
.= (dom fi) +^ (B +^ (C -^ B1)) by ORDINAL3:30 ;
then A8: B +^ (C -^ B1) in dom psi by A2, A6, ORDINAL3:22;
B c= B +^ (C -^ B1) by ORDINAL3:24;
then psi . (B +^ (C -^ B1)) = {} by A2, A4, A6, A7, ORDINAL3:22;
hence (fi ^ psi) . C = 0 by A7, A8, Def1; :: thesis: verum

end;

case A <> 0 ; :: thesis: for b₁, b₂ being set holds
( not b₁ in A or not A in b₂ or ex b₃ being set st
( b₃ in dom (fi ^ psi) & ( for b₄ being set holds
( not b₃ c= b₄ or not b₄ in dom (fi ^ psi) or ( b₁ in (fi ^ psi) . b₄ & (fi ^ psi) . b₄ in b₂ ) ) ) ) )

let B, C be Ordinal; :: thesis: ( not B in A or not A in C or ex b₁ being set st
( b₁ in dom (fi ^ psi) & ( for b₂ being set holds
( not b₁ c= b₂ or not b₂ in dom (fi ^ psi) or ( B in (fi ^ psi) . b₂ & (fi ^ psi) . b₂ in C ) ) ) ) )
assume that
A9: B in A and
A10: A in C ; :: thesis: ex b₁ being set st
( b₁ in dom (fi ^ psi) & ( for b₂ being set holds
( not b₁ c= b₂ or not b₂ in dom (fi ^ psi) or ( B in (fi ^ psi) . b₂ & (fi ^ psi) . b₂ in C ) ) ) )
consider D being Ordinal such that
A11: D in dom psi and
A12: for E being Ordinal st D c= E & E in dom psi holds
( B in psi . E & psi . E in C ) by A1, A9, A10;
take D1 = (dom fi) +^ D; :: thesis: ( D1 in dom (fi ^ psi) & ( for b₁ being set holds
( not D1 c= b₁ or not b₁ in dom (fi ^ psi) or ( B in (fi ^ psi) . b₁ & (fi ^ psi) . b₁ in C ) ) ) )
thus D1 in dom (fi ^ psi) by A2, A11, ORDINAL2:32; :: thesis: for b₁ being set holds
( not D1 c= b₁ or not b₁ in dom (fi ^ psi) or ( B in (fi ^ psi) . b₁ & (fi ^ psi) . b₁ in C ) )
let E be Ordinal; :: thesis: ( not D1 c= E or not E in dom (fi ^ psi) or ( B in (fi ^ psi) . E & (fi ^ psi) . E in C ) )
assume that
A13: D1 c= E and
A14: E in dom (fi ^ psi) ; :: thesis: ( B in (fi ^ psi) . E & (fi ^ psi) . E in C )
A15: D c= D +^ (E -^ D1) by ORDINAL3:24;
A16: E = D1 +^ (E -^ D1) by A13, ORDINAL3:def 5
.= (dom fi) +^ (D +^ (E -^ D1)) by ORDINAL3:30 ;
then A17: D +^ (E -^ D1) in dom psi by A2, A14, ORDINAL3:22;
then (fi ^ psi) . E = psi . (D +^ (E -^ D1)) by A16, Def1;
hence ( B in (fi ^ psi) . E & (fi ^ psi) . E in C ) by A12, A15, A17; :: thesis: verum

end;

end;