reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;

theorem Th7:
  u1 = p1 & u2 = p2 implies (Pitag_dist 2).(u1,u2) = sqrt ((p1`1 -
  p2`1)^2 + (p1`2 - p2`2)^2)
proof
  assume that
A1: u1 = p1 and
A2: u2 = p2;
  reconsider v1 = u1, v2 = u2 as Element of REAL 2;
  reconsider p11 = p1`1, p21 = p2`1, p12 = p1`2, p22 = p2`2 as Element of REAL
     by XREAL_0:def 1;
  p1 = |[p1`1,p1`2]| & u2 = <* p2`1,p2`2 *> by A2,EUCLID:53;
  then v1-v2= <* diffreal.(p11,p21), diffreal.(p12,p22) *> by A1,FINSEQ_2:75
    .= <* p1`1 - p2`1, diffreal.(p1`2,p2`2) *> by BINOP_2:def 10
    .= <* p1`1 - p2`1,p1`2 - p2`2*> by BINOP_2:def 10;
  then abs (v1-v2) = <* absreal.(p1`1-p2`1),absreal.(p1`2-p2`2) *> by
FINSEQ_2:36
    .= <* |. p1`1-p2`1 .|,absreal.(p1`2-p2`2) *> by EUCLID:def 2
    .= <* |. p1`1-p2`1 .|,|. p1`2-p2`2.| *> by EUCLID:def 2;
  then
A3: sqr abs (v1-v2) = <* sqrreal.(|. p1`1 - p2`1 .|),sqrreal.(|. p1`2 -
  p2`2 .|) *> by FINSEQ_2:36
    .= <* (|. p1`1 - p2`1 .|)^2,sqrreal.(|. p1`2-p2`2 .|) *> by RVSUM_1:def 2
    .= <* (|. p1`1 - p2`1 .|)^2,(|.p1`2-p2`2.|)^2 *> by RVSUM_1:def 2
    .= <* (p1`1 - p2`1)^2,(|.p1`2-p2`2.|)^2 *> by COMPLEX1:75
    .= <* (p1`1-p2`1)^2,(p1`2 - p2`2)^2 *> by COMPLEX1:75;
  (Pitag_dist 2).(u1,u2) = |.v1 - v2.| by EUCLID:def 6
    .= sqrt Sum sqr abs (v1-v2) by EUCLID:25;
  hence thesis by A3,RVSUM_1:77;
end;
