reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem
  sqrt(a^2+b^2) <= |.a.|+|.b.|
proof
A1: (sqrt(a^2+b^2))^2>=0 by XREAL_1:63;
  a^2>=0 & b^2>=0 by XREAL_1:63;
  then
A2: (sqrt(a^2+b^2))^2=a^2+b^2 by SQUARE_1:def 2;
  (|.a.|+|.b.|)^2=(|.a.|)^2+2*(|.a.|)*(|.b.|)+(|.b.|)^2 & |.a.|^2=a^2
           by Th75;
  then
  (|.a.|+|.b.|)^2 - (sqrt(a^2+b^2))^2= a^2+2*(|.a.|)*(|.b.|)+b
  ^2-(a^2+b^2) by A2,Th75
    .=2*(|.a.|)*(|.b.|);
  then (|.a.|+|.b.|)^2>= (sqrt(a^2+b^2))^2 by XREAL_1:49;
  then sqrt((|.a.|+|.b.|)^2)>=sqrt((sqrt(a^2+b^2))^2) by A1,SQUARE_1:26;
  hence thesis by A2,SQUARE_1:22;
end;
