reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th25:
  (-1)(#)cos is_integral_of sin,REAL
proof
A1: dom(sin|REAL) = REAL /\ REAL by SIN_COS:24;
A2: dom((-1)(#)cos) =[#]REAL by SIN_COS:24,VALUED_1:def 5;
A3: now
    let x be object;
    assume
A4: x in dom(((-1)(#)cos)`|REAL);
    then reconsider z=x as Real;
    (((-1)(#)cos)`|REAL).x=(-1)*diff(cos,z) by A2,A4,FDIFF_1:20,SIN_COS:67;
    then (((-1)(#)cos)`|REAL).x=(-1)*(-sin.z) by SIN_COS:63;
    hence (((-1)(#)cos)`|REAL).x=(sin|REAL).x;
  end;
A5: (-1)(#)cos is_differentiable_on REAL by A2,FDIFF_1:20,SIN_COS:67;
  then dom (((-1)(#)cos)`|REAL) = REAL by FDIFF_1:def 7;
  then ((-1)(#)cos)`|REAL=sin|REAL by A1,A3,FUNCT_1:2;
  hence thesis by A5,Lm1;
end;
