theorem
  for x be Real st cos.x <> 0 holds tan.x = tan x
proof
  let x be Real;
  assume
A1: cos.x <> 0;
A2: x in REAL by XREAL_0:def 1;
  not x in cos"{0}
  proof
    assume x in cos"{0};
    then cos.x in {0} by FUNCT_1:def 7;
    hence contradiction by A1,TARSKI:def 1;
  end;
  then x in dom cos \ cos"{0} by SIN_COS:24,XBOOLE_0:def 5,A2;
  then x in dom sin /\ (dom cos \ cos"{0}) by SIN_COS:24,XBOOLE_0:def 4,A2;
  then x in dom (sin/cos) by RFUNCT_1:def 1;
  then tan.x = (sin x)/(cos x) by RFUNCT_1:def 1
    .= tan x by SIN_COS4:def 1;
  hence thesis;
end;
