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 Th12:
  cos(x/2) >=0 implies cos(x/2)=sqrt((1+cos x)/2)
proof
  assume
A1: cos(x/2) >=0;
  sqrt((1+cos(x))/2)=sqrt((1+cos(2*(x/2)))/2)
    .=sqrt((1+(2*(cos(x/2))^2-1))/2) by SIN_COS5:7
    .=|.cos(x/2).| by COMPLEX1:72
    .= cos(x/2) by A1,ABSVALUE:def 1;
  hence thesis;
end;
