 reserve n,m,i,p for Nat,
         h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
 reserve f,f1,f2,g for Function of REAL,REAL;

theorem
  x0 in dom tan & x1 in dom tan implies
  [!tan(#)tan(#)cos,x0,x1!] = [!tan(#)sin,x0,x1!]
proof
  assume
A1:x0 in dom tan & x1 in dom tan;
  [!tan(#)tan(#)cos,x0,x1!] = ((tan(#)tan).x0*cos.x0
       -(tan(#)tan(#)cos).x1)/(x0-x1) by VALUED_1:5
    .= (tan.x0*tan.x0*cos.x0-(tan(#)tan(#)cos).x1)/(x0-x1) by VALUED_1:5
    .= (tan.x0*tan.x0*cos.x0-(tan(#)tan).x1*cos.x1)/(x0-x1) by VALUED_1:5
    .= (tan.x0*tan.x0*cos.x0-tan.x1*tan.x1*cos.x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0*(cos.x0)")*tan.x0*cos.x0-tan.x1*tan.x1*cos.x1)/(x0-x1)
                                                   by A1,RFUNCT_1:def 1
    .= ((sin.x0*(cos.x0)"*tan.x0*cos.x0)
       -(sin.x1*(cos.x1)"*tan.x1*cos.x1))/(x0-x1) by A1,RFUNCT_1:def 1
    .= ((sin.x0*(cos.x0*(1/cos.x0))*tan.x0)
       -(sin.x1*(cos.x1*(1/cos.x1))*tan.x1))/(x0-x1)
    .= ((sin.x0*1*tan.x0)
       -(sin.x1*(cos.x1*(1/cos.x1))*tan.x1))/(x0-x1)
                                          by A1,FDIFF_8:1,XCMPLX_1:106
    .= ((sin.x0*1*tan.x0)
       -(sin.x1*1*tan.x1))/(x0-x1) by A1,FDIFF_8:1,XCMPLX_1:106
    .= ((tan(#)sin).x0-tan.x1*sin.x1)/(x0-x1) by VALUED_1:5
    .= [!tan(#)sin,x0,x1!] by VALUED_1:5;
  hence thesis;
end;
