let fi be Ordinal-Sequence; :: thesis: for A, B being Ordinal st A is_limes_of fi holds
B +^ A is_limes_of B +^ fi
let A, B be Ordinal; :: thesis: ( A is_limes_of fi implies B +^ A is_limes_of B +^ fi )
assume A1: ( ( A = 0 & ex B being Ordinal st
( B in dom fi & ( for C being Ordinal st B c= C & C in dom fi holds
fi . 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 fi & ( for E being Ordinal st D c= E & E in dom fi holds
( B in fi . E & fi . E in C ) ) ) ) ) ) ; :: according to ORDINAL2:def 9 :: thesis: B +^ A is_limes_of B +^ fi
A2: dom fi = dom (B +^ fi) by ORDINAL3:def 1;
per cases ( B +^ A = 0 or B +^ A <> 0 ) ;
:: according to ORDINAL2:def 9
case A3: B +^ A = 0 ; :: thesis: ex b1 being set st
( b1 in dom (B +^ fi) & ( for b2 being set holds
( not b1 c= b2 or not b2 in dom (B +^ fi) or (B +^ fi) . b2 = 0 ) ) )
then consider A1 being Ordinal such that
A4: A1 in dom fi and
A5: for C being Ordinal st A1 c= C & C in dom fi holds
fi . C = {} by A1, ORDINAL3:26;
take A1 ; :: thesis: ( A1 in dom (B +^ fi) & ( for b1 being set holds
( not A1 c= b1 or not b1 in dom (B +^ fi) or (B +^ fi) . b1 = 0 ) ) )
thus A1 in dom (B +^ fi) by A4, ORDINAL3:def 1; :: thesis: for b1 being set holds
( not A1 c= b1 or not b1 in dom (B +^ fi) or (B +^ fi) . b1 = 0 )
let C be Ordinal; :: thesis: ( not A1 c= C or not C in dom (B +^ fi) or (B +^ fi) . C = 0 )
assume that
A6: A1 c= C and
A7: C in dom (B +^ fi) ; :: thesis: (B +^ fi) . C = 0
A8: (B +^ fi) . C = B +^ (fi . C) by A2, A7, ORDINAL3:def 1;
fi . C = {} by A2, A5, A6, A7;
hence (B +^ fi) . C = 0 by A3, A8, ORDINAL3:26; :: thesis: verum
end;
case B +^ A <> 0 ; :: thesis: for b1, b2 being set holds
( not b1 in B +^ A or not B +^ A in b2 or ex b3 being set st
( b3 in dom (B +^ fi) & ( for b4 being set holds
( not b3 c= b4 or not b4 in dom (B +^ fi) or ( b1 in (B +^ fi) . b4 & (B +^ fi) . b4 in b2 ) ) ) ) )
now :: thesis: for B1, B2 being Ordinal st B1 in B +^ A & B +^ A in B2 holds
ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
per cases ( A = {} or A <> {} ) ;
suppose A9: A = {} ; :: thesis: for B1, B2 being Ordinal st B1 in B +^ A & B +^ A in B2 holds
ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
then consider A1 being Ordinal such that
A10: A1 in dom fi and
A11: for C being Ordinal st A1 c= C & C in dom fi holds
fi . C = {} by A1;
let B1, B2 be Ordinal; :: thesis: ( B1 in B +^ A & B +^ A in B2 implies ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) ) )
assume that
A12: B1 in B +^ A and
A13: B +^ A in B2 ; :: thesis: ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
take A1 = A1; :: thesis: ( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
thus A1 in dom (B +^ fi) by A10, ORDINAL3:def 1; :: thesis: for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 )
let C be Ordinal; :: thesis: ( A1 c= C & C in dom (B +^ fi) implies ( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) )
assume that
A14: A1 c= C and
A15: C in dom (B +^ fi) ; :: thesis: ( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 )
(B +^ fi) . C = B +^ (fi . C) by A2, A15, ORDINAL3:def 1;
hence ( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) by A2, A9, A11, A12, A13, A14, A15; :: thesis: verum
end;
suppose A16: A <> {} ; :: thesis: for B1, B2 being Ordinal st B1 in B +^ A & B +^ A in B2 holds
ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
let B1, B2 be Ordinal; :: thesis: ( B1 in B +^ A & B +^ A in B2 implies ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) ) )
assume that
A17: B1 in B +^ A and
A18: B +^ A in B2 ; :: thesis: ex A1 being Ordinal st
( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
B1 -^ B in A by A16, A17, ORDINAL3:60;
then consider A1 being Ordinal such that
A19: A1 in dom fi and
A20: for C being Ordinal st A1 c= C & C in dom fi holds
( B1 -^ B in fi . C & fi . C in B2 -^ B ) by A1, A18, ORDINAL3:61;
A21: B1 c= B +^ (B1 -^ B) by ORDINAL3:62;
A22: B c= B +^ A by ORDINAL3:24;
B +^ A c= B2 by A18, ORDINAL1:def 2;
then B c= B2 by A22;
then A23: B +^ (B2 -^ B) = B2 by ORDINAL3:def 5;
take A1 = A1; :: thesis: ( A1 in dom (B +^ fi) & ( for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) ) )
thus A1 in dom (B +^ fi) by A19, ORDINAL3:def 1; :: thesis: for C being Ordinal st A1 c= C & C in dom (B +^ fi) holds
( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 )
let C be Ordinal; :: thesis: ( A1 c= C & C in dom (B +^ fi) implies ( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) )
assume that
A24: A1 c= C and
A25: C in dom (B +^ fi) ; :: thesis: ( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 )
A26: (B +^ fi) . C = B +^ (fi . C) by A2, A25, ORDINAL3:def 1;
reconsider E = fi . C as Ordinal ;
B1 -^ B in E by A2, A20, A24, A25;
then A27: B +^ (B1 -^ B) in B +^ E by ORDINAL2:32;
E in B2 -^ B by A2, A20, A24, A25;
hence ( B1 in (B +^ fi) . C & (B +^ fi) . C in B2 ) by A21, A26, A23, A27, ORDINAL1:12, ORDINAL2:32; :: thesis: verum
end;
end;
end;
hence for b1, b2 being set holds
( not b1 in B +^ A or not B +^ A in b2 or ex b3 being set st
( b3 in dom (B +^ fi) & ( for b4 being set holds
( not b3 c= b4 or not b4 in dom (B +^ fi) or ( b1 in (B +^ fi) . b4 & (B +^ fi) . b4 in b2 ) ) ) ) ) ; :: thesis: verum
end;
end;