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 Th6:
  sin(x/2) < 0 implies sin(x/2)=-sqrt((1-cos(x))/2)
proof
  assume
A1: sin(x/2) < 0;
  sqrt((1-cos(x))/2)=sqrt((1-cos(2*(x/2)))/2)
    .=sqrt((1-(1-2*(sin(x/2))^2))/2) by SIN_COS5:7
    .=|.sin(x/2).| by COMPLEX1:72
    .=-sin(x/2) by A1,ABSVALUE:def 1;
  hence thesis;
end;
