reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  (for x holds f.x = (tan(#)cos).x) & x0 in dom tan & x1 in dom tan
  implies [!f,x0,x1!] = (sin(x0)-sin(x1))/(x0-x1)
proof
  assume that
A1:for x holds f.x = (tan(#)cos).x and
A2:x0 in dom tan & x1 in dom tan;
A3:f.(x0) = (tan(#)cos).x0 by A1;
f.(x1) = (tan(#)cos).x1 by A1;
  then [!f,x0,x1!] = ((tan.(x0))*(cos.(x0)) - (tan(#)cos).x1)/(x0-x1)
                                             by A3,VALUED_1:5
    .= ((tan.(x0))*(cos.(x0))-(tan.(x1))*(cos.(x1)))/(x0-x1) by VALUED_1:5
    .= ((sin.(x0)*(cos.(x0))")*(cos.(x0))-(tan.(x1))*(cos.(x1)))/(x0-x1)
                                                     by A2,RFUNCT_1:def 1
    .= (sin(x0)/cos(x0)*cos(x0)-sin(x1)/cos(x1)*cos(x1))/(x0-x1)
                                                by A2,RFUNCT_1:def 1
    .= (sin(x0)/(cos(x0)/cos(x0))-sin(x1)/cos(x1)*cos(x1))/(x0-x1)
                                                            by XCMPLX_1:82
    .= (sin(x0)/(cos(x0)*(1/cos(x0)))-sin(x1)/(cos(x1)/cos(x1)))/(x0-x1)
                                                          by XCMPLX_1:82
    .= (sin(x0)/1-sin(x1)/(cos(x1)*(1/cos(x1))))/(x0-x1)
                                              by A2,FDIFF_8:1,XCMPLX_1:106
    .= (sin(x0)/1-sin(x1)/1)/(x0-x1) by A2,FDIFF_8:1,XCMPLX_1:106
    .= (sin(x0)-sin(x1))/(x0-x1);
  hence thesis;
end;
