reserve A,B,C,D,p,q,r for Element of LTLB_WFF,
        F,G,X for Subset of LTLB_WFF,
        M for LTLModel,
        i,j,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;

theorem th263: :: 2.6.3
  F\/{A} |=0 B iff F|=0 A=>B
  proof
    hereby assume A3:F\/{A} |=0 B;
      thus F|=0 A=>B
      proof
        let M;
        assume
A4:     M |=0 F;
A2:     (SAT M).[0,A=>B]= (SAT M).[0,A] => (SAT M).[0,B]
        by LTLAXIO1:def 11;
        thus M |=0 A=> B
        proof
          per cases by XBOOLEAN:def 3;
          suppose (SAT M).[0,A] = 0;
            hence (SAT M).[0,A=>B]= 1 by A2;
          end;
          suppose (SAT M).[0,A] = 1;then
            M |=0 A;then
            M |=0 {A} by th263pb;then
            M |=0 B by A3,th263pa,A4;
            hence (SAT M).[0,A=>B]= 1 by A2;
          end;
        end;
      end;
    end;
    assume
A6: F|=0 A=>B;
    let M;
    assume M |=0 F\/{A};then
A5: M |=0 F & M |=0 {A} by th263pa;then
A7: M |=0 A by th263pb;
    M |=0 A=>B by A5,A6;then
    (SAT M).[0,A] => (SAT M).[0,B] = 1 by LTLAXIO1:def 11;
    hence M |=0 B by A7;
end;
