theorem
  a*b+b*c+c*a=1 implies a+b+c>=sqrt 3
proof
  assume
A1: a*b+b*c+c*a=1;
  then a^2+b^2+c^2>=1 by SERIES_3:10;
  then a^2+b^2+c^2+2>=1+2 by XREAL_1:6;
  then sqrt((a+b+c)^2)>=sqrt(3) by A1,SQUARE_1:26;
  hence thesis by SQUARE_1:22;
end;
