let fi be Ordinal-Sequence; :: thesis: for C being Ordinal st ( for A being Ordinal st A in dom fi holds
fi . A = C +^ A ) holds
fi is increasing
let C be Ordinal; :: thesis: ( ( for A being Ordinal st A in dom fi holds
fi . A = C +^ A ) implies fi is increasing )
assume A1: for A being Ordinal st A in dom fi holds
fi . A = C +^ A ; :: thesis: fi is increasing
let A be Ordinal; :: according to ORDINAL2:def 12 :: thesis: for b₁ being set holds
( not A in b₁ or not b₁ in dom fi or fi . A in fi . b₁ )
let B be Ordinal; :: thesis: ( not A in B or not B in dom fi or fi . A in fi . B )
assume that
A2: A in B and
A3: B in dom fi ; :: thesis: fi . A in fi . B
A4: fi . B = C +^ B by A1, A3;
fi . A = C +^ A by A1, A2, A3, ORDINAL1:10;
hence fi . A in fi . B by A2, A4, ORDINAL2:32; :: thesis: verum