reserve Z for open Subset of REAL;

theorem Th12:
  for M,L be Real st M >= 0 & L >= 0
  for e be Real st e > 0
   ex n be Nat st for m be Nat st n <= m holds (M*L |^ m / (m!)) < e
proof
  let M,L be Real such that
A1: M >=0 and
A2: L >= 0;
A3: L rExpSeq is summable by SIN_COS:45;
  then
A4: M(#)(L rExpSeq) is convergent by SEQ_2:7,SERIES_1:4;
  lim (L rExpSeq)=0 by A3,SERIES_1:4;
  then
A5: lim (M(#)(L rExpSeq))=M*0 by A3,SEQ_2:8,SERIES_1:4;
  let p be Real;
  assume p > 0;
  then consider n be Nat such that
A6: for m be Nat st n<=m holds |.(M(#)(L rExpSeq)).m - 0.|
  < p by A4,A5,SEQ_2:def 7;
  take n;
A7: for n be Element of NAT holds M*(L |^ n /(n!))=(M(#)(L rExpSeq)).n
  proof
    let n be Element of NAT;
    M*(L |^ n /(n!)) = M*((L rExpSeq).n) by SIN_COS:def 5
      .= (M(#)(L rExpSeq)).n by SEQ_1:9;
    hence thesis;
  end;
  now
    let m be Nat such that
A8: n <= m;
A9: m in NAT by ORDINAL1:def 12;
A10: L |^ m >= 0 & m! > 0 by A2,POWER:3;
    |.(M(#)(L rExpSeq)).m - 0.| = |.M*(L |^ m /(m!)).| by A7,A9
      .=|.M.|*|.L |^ m /(m!).| by COMPLEX1:65
      .=M*|.L |^ m /(m!).| by A1,ABSVALUE:def 1
      .=M*(L |^ m /(m!)) by A10,ABSVALUE:def 1
      .=M*L |^ m /(m!) by XCMPLX_1:74;
    hence (M*L |^ m /(m!)) < p by A6,A8;
  end;
  hence thesis;
end;
