reserve A,B,p,q,r for Element of LTLB_WFF,
  M for LTLModel,
  j,k,n for Element of NAT,
  i for Nat,
  X for Subset of LTLB_WFF,
  F for finite Subset of LTLB_WFF,
  f for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y,z for set,
  P,Q,R for PNPair;

theorem Th10:
  for F,G be Subset of [:LTLB_WFF**,LTLB_WFF**:] holds (F \/ G)^ = F^ \/ G^
  proof
    let F,G be Subset of [:l**,l**:];
    hereby
      let x be object;
      assume x in (F \/ G)^;
      then consider P such that
A1:   x = P^ and
A2:   P in F \/ G;
      P in F or P in G by A2,XBOOLE_0:def 3;
      then x in F^ or x in G^ by A1;
      hence x in F^ \/ G^ by XBOOLE_0:def 3;
    end;
    let x be object;
    assume
A3: x in F^ \/ G^;
    per cases by A3,XBOOLE_0:def 3;
    suppose x in F^;
      then consider P such that
A4:   x = P^ and
A5:   P in F;
      P in F \/ G by A5,XBOOLE_0:def 3;
      hence x in (F \/ G)^ by A4;
    end;
    suppose x in G^;
      then consider P such that
A6:   x = P^ and
A7:   P in G;
      P in F \/ G by A7,XBOOLE_0:def 3;
      hence x in (F \/ G)^ by A6;
    end;
  end;
