reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem
  cos(a-b) = cos(a)*cos(b)+sin(a)*sin(b)
proof
  thus cos(a-b) =(cos.a) *(cos.(-b))-(sin.(a)) * (sin.(-b)) by Th73
    .=(cos.a) *(cos.b)-(sin.a) * (sin.(-b)) by Th30
    .=(cos.a) *(cos.b)-(sin.a) * -(sin.b) by Th30
    .=(cos a) *(cos b)+(sin a) * (sin b);
end;
