
theorem Th25:
  for L being LATTICE holds L is distributive iff L opp is distributive
proof
  let L be LATTICE;
  hereby
    assume
A1: L is distributive;
    thus L opp is distributive
    proof
      let x,y,z be Element of L opp;
      thus x "/\" (y "\/" z) = (~x)"\/"~(y "\/" z) by Th24
        .= (~x)"\/"((~y)"/\"~z) by Th22
        .= ((~x)"\/"(~y))"/\"((~x)"\/"~z) by A1,WAYBEL_1:5
        .= (~(x"/\"y))"/\"((~x)"\/"~z) by Th24
        .= (~(x"/\"y))"/\"~(x"/\"z) by Th24
        .= (x "/\" y) "\/" (x "/\" z) by Th22;
    end;
  end;
  assume
A2: L opp is distributive;
  let x,y,z be Element of L;
  thus x "/\" (y "\/" z) = (x~)"\/"((y "\/" z)~) by Th21
    .= (x~)"\/"((y~)"/\"(z~)) by Th23
    .= ((x~)"\/"(y~))"/\"((x~)"\/"(z~)) by A2,WAYBEL_1:5
    .= ((x"/\"y)~)"/\"((x~)"\/"(z~)) by Th21
    .= ((x"/\"y)~)"/\"((x"/\"z)~) by Th21
    .= (x "/\" y) "\/" (x "/\" z) by Th23;
end;
