let A be Ordinal; :: thesis: ( A <> {} & A is limit_ordinal implies 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 )
assume A1: ( A <> {} & A is limit_ordinal ) ; :: thesis: 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
let fi be Ordinal-Sequence; :: thesis: ( dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp 1,B ) implies 1 is_limes_of fi )
assume A2: ( dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp 1,B ) ) ; :: thesis: 1 is_limes_of fi
per cases ( 1 = {} or 1 <> {} ) ;
:: according to ORDINAL2:def 13
case 1 = {} ; :: thesis: ex b1 being set st
( b1 in dom fi & ( for b2 being set holds
( not b1 c= b2 or not b2 in dom fi or fi . b2 = {} ) ) )
hence ex b1 being set st
( b1 in dom fi & ( for b2 being set holds
( not b1 c= b2 or not b2 in dom fi or fi . b2 = {} ) ) ) ; :: thesis: verum
end;
case 1 <> {} ; :: thesis: for b1, b2 being set holds
( not b1 in 1 or not 1 in b2 or ex b3 being set st
( b3 in dom fi & ( for b4 being set holds
( not b3 c= b4 or not b4 in dom fi or ( b1 in fi . b4 & fi . b4 in b2 ) ) ) ) )
let A1, A2 be Ordinal; :: thesis: ( not A1 in 1 or not 1 in A2 or ex b1 being set st
( b1 in dom fi & ( for b2 being set holds
( not b1 c= b2 or not b2 in dom fi or ( A1 in fi . b2 & fi . b2 in A2 ) ) ) ) )
assume A3: ( A1 in 1 & 1 in A2 ) ; :: thesis: ex b1 being set st
( b1 in dom fi & ( for b2 being set holds
( not b1 c= b2 or not b2 in dom fi or ( A1 in fi . b2 & fi . b2 in A2 ) ) ) )
take B = {} ; :: thesis: ( B in dom fi & ( for b1 being set holds
( not B c= b1 or not b1 in dom fi or ( A1 in fi . b1 & fi . b1 in A2 ) ) ) )
thus B in dom fi by A1, A2, ORDINAL3:10; :: thesis: for b1 being set holds
( not B c= b1 or not b1 in dom fi or ( A1 in fi . b1 & fi . b1 in A2 ) )
let D be Ordinal; :: thesis: ( not B c= D or not D in dom fi or ( A1 in fi . D & fi . D in A2 ) )
assume ( B c= D & D in dom fi ) ; :: thesis: ( A1 in fi . D & fi . D in A2 )
then ( exp 1,D = fi . D & exp 1,D = 1 ) by A2, ORDINAL2:63;
hence ( A1 in fi . D & fi . D in A2 ) by A3; :: thesis: verum
end;
end;