reserve x,x0, r,r1,r2 for Real,
      th for Real,

  rr for set,

  rseq for Real_Sequence;

theorem Th29:
  sin.(PI/4) = 1/sqrt 2 & cos.(PI/4) = 1/sqrt 2
proof
A1: (sqrt(1/2))^2 = 1/2 by SQUARE_1:def 2;
  1 = (sin.(PI/4))^2 + (sin.(PI/4))^2 by SIN_COS:28,73
    .= 2*(sin.(PI/4))^2;
  then
A2: sin.(PI/4) = sqrt(1/2) or sin.(PI/4) = -sqrt(1/2) by A1,SQUARE_1:40;
  PI/4 < PI/1 by XREAL_1:76;
  then
A3: PI/4 in ].0,PI.[;
  sqrt(1/2) > 0 by SQUARE_1:25;
  hence thesis by A2,A3,COMPTRIG:7,SIN_COS:73,SQUARE_1:32;
end;
