
theorem Th22:
  for S be non empty finite set,
  s be Element of S*,
  f,g be Function of S,BOOLEAN
  holds Coim((f '&' g)*s,TRUE) = Coim (f*s,TRUE) /\ Coim(g*s,TRUE)
  proof
    let S be non empty finite set,
    s be Element of S*,
    f,g be Function of S,BOOLEAN;
    A1: now let x be object;
    A2: dom f = S & dom g = S by FUNCT_2:def 1;
    A3:x in dom ((f '&' g))
    iff x in (dom f /\ dom g) by MARGREL1:def 18;
    A4:x in (f '&' g)"{TRUE} iff x in dom ((f '&' g))
    & (f '&' g).x in {TRUE} by FUNCT_1:def 7;
    x in dom ((f '&' g))
    & (f '&' g).x =TRUE iff
    x in (dom f /\ dom g) & (f.x = TRUE & g.x =TRUE) by Lm3,A3; then
    x in (f '&' g)"{TRUE} iff ((x in dom f ) & (f.x in {TRUE}))
    & ((x in dom g ) & (g.x = TRUE)) by A4,A2,TARSKI:def 1; then
    x in (f '&' g)"{TRUE} iff x in f"{TRUE}
    & ((x in dom g ) & (g.x in {TRUE}))
    by FUNCT_1:def 7,TARSKI:def 1; then
    x in (f '&' g)"{TRUE} iff x in f"{TRUE}
    & x in g"{TRUE} by FUNCT_1:def 7;
    hence x in (f '&' g)"{TRUE} iff x in f"{TRUE} /\ g"{TRUE}
    by XBOOLE_0:def 4;
  end;
  A5:s"(f"{TRUE} /\ g"{TRUE}) c= s" (f"{TRUE}) /\ s"(g"{TRUE}) by RELAT_1:141;
  for x be object st x in s" (f"{TRUE}) /\ s"(g"{TRUE}) holds
  x in s"(f"{TRUE} /\ g"{TRUE})
  proof
    let x be object;
    assume A6:x in s" (f"{TRUE}) /\ s"(g"{TRUE});
    assume A7: not x in s"(f"{TRUE} /\ g"{TRUE});
    x in s"(f"{TRUE}) & x in s"(g"{TRUE}) by A6,XBOOLE_0:def 4;then
    A8:x in dom s & s.x in (f"{TRUE}) & s.x in (g"{TRUE})
    by FUNCT_1:def 7;
    then reconsider y=s.x as Element of S;
    not y in (f"{TRUE} /\ g"{TRUE}) by A7,A8,FUNCT_1:def 7;
    hence contradiction by A8,XBOOLE_0:def 4;
  end;
  then
  A9:s" (f"{TRUE}) /\ s"(g"{TRUE})
  c= s"(f"{TRUE} /\ g"{TRUE}) by TARSKI:def 3;
  thus Coim((f '&' g)*s,TRUE) = s"((f '&' g)"{TRUE}) by RELAT_1:146
  .= s"(f"{TRUE} /\ g"{TRUE}) by A1,TARSKI:2
  .= s" (f"{TRUE}) /\ s"(g"{TRUE}) by A9,A5,XBOOLE_0:def 10
  .= (f*s)"{TRUE} /\ s"(g"{TRUE}) by RELAT_1:146
  .= Coim (f*s,TRUE) /\ Coim(g*s,TRUE) by RELAT_1:146;
end;
