reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th37:
  for p1, p2 being Point of TOP-REAL 3, u1, u2 being Point of Euclid 3 holds
  u1 = p1 & u2 = p2 implies
  (Pitag_dist 3).(u1,u2) =
  sqrt ((p1`1 - p2`1)^2 + (p1`2 - p2`2)^2 + (p1`3 - p2`3)^2)
proof
  let p1, p2 be Point of TOP-REAL 3, u1, u2 be Point of Euclid 3;
  assume
A1: u1 = p1 & u2 = p2;
A2: p1 = |[p1`1,p1`2,p1`3]| by EUCLID_5:3;
   reconsider p21 =  p2`1, p22 = p2`2, p23 = p2`3 as Element of REAL
          by XREAL_0:def 1;
   reconsider p11 =  p1`1, p12 = p1`2, p13 = p1`3 as Element of REAL
          by XREAL_0:def 1;
A3: u2 = <* p21,p22,p23 *> by A1,EUCLID_5:3;
  reconsider v1 = u1, v2 = u2 as Element of REAL 3;
 reconsider d1 = diffreal.(p11,p21), d2 = diffreal.(p12,p22),
       d3 = diffreal.(p13,p23) as Element of REAL;
  v1-v2= <* d1, d2, d3 *> by A1,A2,A3,FINSEQ_2:76
  .= <* p1`1-p2`1, diffreal.(p1`2,p2`2), diffreal.(p1`3,p2`3) *>
  by BINOP_2:def 10
  .= <* p1`1-p2`1, p1`2-p2`2, diffreal.(p1`3,p2`3) *> by BINOP_2:def 10
  .= <* p1`1-p2`1, p1`2-p2`2, p1`3-p2`3 *> by BINOP_2:def 10;
  then abs(v1-v2) = <* absreal.(p1`1-p2`1),absreal.(p1`2-p2`2),
  absreal.(p1`3-p2`3) *> by FINSEQ_2:37
  .= <* |.p1`1-p2`1.|,absreal.(p1`2-p2`2),absreal.(p1`3-p2`3) *>
  by EUCLID:def 2
  .= <* |.p1`1-p2`1.|,|.p1`2-p2`2.|,absreal.(p1`3-p2`3) *> by EUCLID:def 2
  .= <* |.p1`1-p2`1.|,|.p1`2-p2`2.|,|.p1`3-p2`3.| *> by EUCLID:def 2;
  then
A4: sqr abs (v1-v2) = <* sqrreal.(|.p1`1-p2`1.|),sqrreal.(|.p1`2-p2`2.|),
    sqrreal. |.p1`3-p2`3.| *> by FINSEQ_2:37
    .= <* (|.p1`1-p2`1.|)^2,sqrreal.(|.p1`2-p2`2.|),
    sqrreal.(|.p1`3-p2`3.|) *> by RVSUM_1:def 2
    .= <* (|.p1`1-p2`1.|)^2,(|.p1`2-p2`2.|)^2,sqrreal.(|.p1`3-p2`3.|) *>
    by RVSUM_1:def 2
    .= <* (|.p1`1-p2`1.|)^2,(|.p1`2-p2`2.|)^2,(|.p1`3-p2`3.|)^2 *>
    by RVSUM_1:def 2
    .= <* (p1`1-p2`1)^2,(|.p1`2-p2`2.|)^2,(|.p1`3-p2`3.|)^2 *>
    by COMPLEX1:75
    .= <* (p1`1-p2`1)^2,(p1`2-p2`2)^2,(|.p1`3-p2`3.|)^2 *> by COMPLEX1:75
    .= <* (p1`1-p2`1)^2,(p1`2-p2`2)^2,(p1`3-p2`3)^2 *> by COMPLEX1:75;
  (Pitag_dist 3).(u1,u2) = |.v1 - v2.| by EUCLID:def 6
    .= sqrt Sum sqr abs (v1-v2) by EUCLID:25;
  hence thesis by A4,RVSUM_1:78;
end;
