reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th8:
  sin(-PI/4) = -sqrt(2)/2
proof
A1: cos(-PI/2) = sin(PI/2-(-PI/2))by SIN_COS:79
    .= 0 by SIN_COS:77;
A2: sqrt 2 > 0 by SQUARE_1:25;
  sin((-PI/2)/2)=-sqrt((1-cos(-PI/2))/2) by Lm2,Th6;
  then sin(-PI/4) = -1 /sqrt 2 by A1,SQUARE_1:18,30
    .= -(sqrt(2) * 1)/(sqrt(2)*sqrt(2)) by A2,XCMPLX_1:91
    .= -(sqrt(2) * 1)/(sqrt(2))^2;
  hence thesis by SQUARE_1:def 2;
end;
