reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;

theorem Th19:
  sqrt(r1*r2)=(r1+r2)/2 iff r1 = r2
   proof
   hereby
     assume
A1:  sqrt(r1*r2)=(r1+r2)/2;
A2:  ((sqrt(r1*r2))*2)^2=(sqrt(r1*r2))^2*2^2.=r1*r2*4 by SQUARE_1:def 2
     .= 4*r1*r2;
     0 =(r1+r2)^2-((sqrt(r1*r2))*2)^2 by A1
     .= (r1-r2)^2 by A2; then
     r1-r2 = 0;
     hence r1=r2;
     end;
     assume
A3:  r1=r2; then
     sqrt(r1*r2) = (sqrt(r1))^2 by SQUARE_1:29
     .= (r1+r2)/2 by A3,SQUARE_1:def 2;
     hence sqrt(r1*r2) = (r1+r2)/2;
   end;
