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 Th23:
  sin is_integral_of cos,REAL
proof
A1: dom(cos|REAL) = REAL /\ REAL by SIN_COS:24;
A2: now
    let x be object;
    assume
A3: x in dom(sin`|REAL);
    then reconsider z=x as Real;
    (sin`|REAL).x = diff(sin,z) by A3,FDIFF_1:def 7,SIN_COS:68;
    then (sin`|REAL).x = cos.z by SIN_COS:64;
    hence (sin`|REAL).x = (cos|REAL).x;
  end;
  dom(sin`|REAL) = REAL by FDIFF_1:def 7,SIN_COS:68;
  then sin`|REAL=cos|REAL by A1,A2,FUNCT_1:2;
  hence thesis by Lm1,SIN_COS:68;
end;
