
theorem Th63:
  for L being non empty transitive RelStr for S being non empty
full SubRelStr of L for X being Subset of S st ex_inf_of X,L & "/\"(X,L) in the
  carrier of S holds ex_inf_of X,S & "/\"(X,S) = "/\"(X,L)
proof
  let L be non empty transitive RelStr;
  let S be non empty full SubRelStr of L;
  let X be Subset of S;
  assume that
A1: ex_inf_of X,L and
A2: "/\"(X,L) in the carrier of S;
  reconsider a = "/\"(X,L) as Element of S by A2;
A3: now
    "/\"(X,L) is_<=_than X by A1,Def10;
    hence a is_<=_than X by Th61;
    let b be Element of S;
    reconsider b9 = b as Element of L by Th58;
    assume b is_<=_than X;
    then b9 is_<=_than X by Th62;
    then b9 <= "/\"(X,L) by A1,Def10;
    hence b <= a by Th60;
  end;
  consider a9 being Element of L such that
A4: X is_>=_than a9 and
A5: for b being Element of L st X is_>=_than b holds b <= a9 and
  for c being Element of L st X is_>=_than c & for b being Element of L st
  X is_>=_than b holds b <= c holds c = a9 by A1;
A6: a9 = "/\"(X,L) by A1,A4,A5,Def10;
  thus ex_inf_of X,S
  proof
    take a;
    thus a is_<=_than X & for b being Element of S st b is_<=_than X holds b
    <= a by A3;
    let c be Element of S;
    reconsider c9 = c as Element of L by Th58;
    assume X is_>=_than c;
    then
A7: X is_>=_than c9 by Th62;
    assume for b being Element of S st X is_>=_than b holds b <= c;
    then
A8: a <= c by A3;
    now
      let b be Element of L;
      assume X is_>=_than b;
      then
A9:   b <= a9 by A5;
      a9 <= c9 by A6,A8,Th59;
      hence b <= c9 by A9,ORDERS_2:3;
    end;
    hence thesis by A1,A7,Def10;
  end;
  hence thesis by A3,Def10;
end;
