reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th53:
  a>0 implies a #Q p * a #Q q = a #Q (p+q)
proof
  set dp = denominator(p);
  set dq = denominator(q);
  set np = numerator(p);
  set nq = numerator(q);
A1: dp >= 1 by RAT_1:11;
  reconsider ddq=dq as Integer;
  reconsider ddp=dp as Integer;
A2: denominator(p+q) >= 1 by RAT_1:11;
A3: dq >= 1 by RAT_1:11;
  then
A4: dp*dq >= 1 by A1,XREAL_1:159;
  p+q = np/dp + q by RAT_1:15
    .= np/dp + nq/dq by RAT_1:15
    .= (np*dq+nq*dp)/(dp*dq) by XCMPLX_1:116;
  then consider n such that
A5: np*dq+nq*dp = numerator(p+q)*n and
A6: dp*dq = denominator(p+q)*n by RAT_1:27;
  reconsider k=n as Integer;
  assume
A7: a>0;
  then
A8: a #Z (nq-np) >= 0 by Th39;
  n>0 by A6;
  then
A9: n>=0+1 by NAT_1:13;
A10: a #Z np > 0 by A7,Th39;
  then
A11: (a #Z np) |^ (dp+dq)>= 0 by Th6;
A12: a #Z (nq-np) > 0 by A7,Th39;
  then
A13: (a #Z (nq-np)) |^ dp >= 0 by Th6;
A14: a #Q (p+q) > 0 by A7,Th52;
A15: a #Z numerator(p+q) > 0 by A7,Th39;
  thus a #Q p * a #Q q = dp -Root (a #Z np) * dq -Root (a #Z (np+(nq-np)))
    .= dp -Root (a #Z np) * dq -Root (a #Z np * a #Z (nq-np)) by A7,Th44
    .= dp -Root (a #Z np)*(dq -Root (a #Z np)*dq -Root (a #Z (nq-np))) by A10
,A8,Th22,RAT_1:11
    .= dp -Root (a #Z np) * dq -Root (a #Z np) * dq -Root (a #Z (nq-np))
    .= (dp*dq) -Root ((a #Z np) |^ (dp+dq))*dq -Root (a #Z (nq-np)) by A10,A3
,A1,Th26
    .= (dp*dq) -Root ((a #Z np) |^ (dp+dq)) * dq -Root (dp -Root ((a #Z (nq-
  np)) |^ dp)) by A12,A1,Lm2
    .= (dp*dq) -Root ((a #Z np) |^ (dp+dq)) * (dp*dq) -Root ((a #Z (nq-np))
  |^ dp) by A3,A1,A13,Th25
    .= (dp*dq) -Root (((a #Z np) |^ (dp+dq)) * ((a #Z (nq-np)) |^ dp)) by A3,A1
,A13,A11,Th22,XREAL_1:159
    .= (dp*dq) -Root (((a #Z np) #Z (ddp+ddq)) * ((a #Z (nq-np)) |^ dp)) by
Th36
    .= (dp*dq) -Root (((a #Z np) #Z (ddp+ddq)) * ((a #Z (nq-np)) #Z ddp)) by
Th36
    .= (dp*dq) -Root ( a #Z (np*(ddp+ddq)) * ((a #Z (nq-np)) #Z ddp) ) by Th45
    .= (dp*dq) -Root ( a #Z (np*(ddp+ddq)) * a #Z ((nq-np)*ddp) ) by Th45
    .= (dp*dq) -Root (a #Z (np*ddp+np*ddq+(nq-np)*ddp) ) by A7,Th44
    .= (denominator(p+q)*n) -Root (a #Z numerator(p+q) #Z k) by A5,A6,Th45
    .= ((denominator(p+q)*n) -Root (a #Z numerator(p+q))) #Z k by A4,A6,A15
,Th46
    .= ((n*denominator(p+q)) -Root (a #Z numerator(p+q))) |^ n by Th36
    .= (n -Root (a #Q (p+q))) |^ n by A9,A2,A15,Th25
    .= a #Q (p+q) by A9,A14,Lm2;
end;
