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 Th73:
  sin.(th1+th2)=(sin.(th1)) *(cos.(th2))+(cos.(th1)) * (sin.(th2)) &
  cos.(th1+th2)=(cos.(th1)) *(cos.(th2))-(sin.(th1)) * (sin.(th2))
proof
  reconsider th1,th2 as Real;
A1: (th1+th2)*<i> = 0+0+(th1+th2)*<i>;
A2: Sum(th1*<i> ExpSeq ) * Sum(th2*<i> ExpSeq )
  =Sum((th1*<i>+th2*<i>) ExpSeq ) by Lm2
    .=cos.(th1+th2)+(sin.(th1+th2))*<i> by A1,Lm3;
 Sum((th1*<i> ExpSeq) ) * Sum((th2*<i> ExpSeq) )
  =(cos.(th1)+(sin.th1)*<i>)* Sum((th2*<i> ExpSeq) ) by Lm3
    .=(cos.(th1)+sin.(th1)*<i>)*(cos.(th2)+(sin.th2)*<i>) by Lm3
    .=((cos.(th1)) *( cos.(th2))-(sin.(th1)) * (sin.(th2))+
  ((sin.(th1)) * (cos.(th2))+(cos.(th1)) * (sin.(th2)))*<i>);
  hence thesis by A2,COMPLEX1:77;
end;
