
theorem
  for x,y being Element of BOOLEAN holds and2c.<*x,y*> = x '&' 'not' y &
and2c.<*x,y*> = and2a.<*y,x*> & and2c.<*x,y*> = nor2a.<*x,y*> & and2c.<*0,0*>=0
  & and2c.<*0,1*>=0 & and2c.<*1,0*>=1 & and2c.<*1,1*>=0
proof
  let x,y be Element of BOOLEAN;
  thus and2c.<*x,y*> = x '&' 'not' y by Def3;
  thus and2c.<*x,y*> = x '&' 'not' y by Def3
    .=and2a.<*y,x*> by TWOSCOMP:def 2;
  thus and2c.<*x,y*> = 'not' ('not' x 'or' 'not' 'not' y) by Def3
    .= nor2a.<*x,y*> by TWOSCOMP:def 11;
  thus and2c.<*0,0*> = FALSE '&' 'not' FALSE by Def3
    .= 0;
  thus and2c.<*0,1*> = FALSE '&' 'not' TRUE by Def3
    .= 0;
  thus and2c.<*1,0*> = TRUE '&' 'not' FALSE by Def3
    .= 1;
  thus and2c.<*1,1*> = TRUE '&' 'not' TRUE by Def3
    .= 0;
end;
