
theorem Th28:
  for A, B being finite Ordinal-Sequence
  st dom A c= dom B & for a being object st a in dom A holds A.a c= B.a
  holds Sum^ A c= Sum^ B
proof
  let A, B be finite Ordinal-Sequence;
  assume that A1: dom A c= dom B and
    A2: for a being object st a in dom A holds A.a c= B.a;
  set a = dom A;
  consider f1 being Ordinal-Sequence such that
    A3: Sum^ A = last f1 & dom f1 = succ dom A & f1.0 = 0 and
    A4: for n being Nat st n in dom A holds f1.(n+1) = f1.n +^ A.n
    by ORDINAL5:def 8;
  consider f2 being Ordinal-Sequence such that
    A5: Sum^(B|a) = last f2 & dom f2 = succ dom(B|a) & f2.0 = 0 and
    A6: for n being Nat st n in dom(B|a) holds f2.(n+1) = f2.n +^ (B|a).n
    by ORDINAL5:def 8;
  defpred P[Nat] means $1 in succ a implies f1.$1 c= f2.$1;
  A7: P[0] by A3;
  A8: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume A9: P[n];
    assume n+1 in succ a;
    then A10: succ n in succ a by Lm5;
    then A11: n in a by ORDINAL3:3;
    n in succ n by ORDINAL1:6;
    then A12: f1.n c= f2.n by A9, A10, ORDINAL1:10;
    A13: f1.(n+1) = f1.n +^ A.n by A4, A10, ORDINAL3:3;
    A14: n in dom(B|a) by A1, A11, RELAT_1:62;
    then A15: f2.(n+1) = f2.n +^ (B|a).n by A6
      .= f2.n +^ B.n by A14, FUNCT_1:47;
    A.n c= B.n by A2, A10, ORDINAL3:3;
    hence thesis by A12, A13, A15, ORDINAL3:18;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A7,A8);
  then f1.a c= f2.a by ORDINAL1:6;
  then last f1 c= f2.a by A3, ORDINAL2:6;
  then last f1 c= f2.dom(B|a) by A1, RELAT_1:62;
  then A17: Sum^ A c= Sum^(B|a) by A3, A5, ORDINAL2:6;
  Sum^(B|a) c= Sum^ B by Th27;
  hence thesis by A17, XBOOLE_1:1;
end;
