reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;

theorem Th71:
  q=<*r*> implies |.q.|=|.r.|
proof
  assume
A1: q=<*r*>;
  reconsider rr=r as Element of REAL by XREAL_0:def 1;
  reconsider xr=<*rr*> as Element of REAL 1;
  reconsider qk = (q/.1)^2 as Element of REAL by XREAL_0:def 1;
  len xr=1 by FINSEQ_1:39;
  then
A2: q/.1=xr.1 by A1,FINSEQ_4:15;
  then len (sqr xr) =1 & (sqr xr).1=(q/.1)^2 by CARD_1:def 7,VALUED_1:11;
  then
A3: sqr xr=<*qk*> by FINSEQ_1:40;
  sqrt (q/.1)^2 =|.q/.1.| by COMPLEX1:72
    .=|.r.| by A2;
   then |.xr.|=|.rr.| by A3,FINSOP_1:11;
  hence thesis by A1;
end;
